July 08, 2025
We investigate the existence of a robust, i.e., continuous, representation of the conditional distribution in a stochastic filtering model for multidimensional correlated jump-diffusions. Even in the absence of jumps, it is known that in general such a representation can only be continuous with respect to rough path topologies, leading us naturally to express the conditional dynamics as a rough stochastic differential equation with jumps. Via the analysis of such equations, including exponential moments, Skorokhod continuity, and randomisation of the rough path, we establish several novel robustness results for stochastic filters.
Keywords: càdlàg rough paths, rough stochastic differential equations, John–Nirenberg inequality, robust stochastic filtering.
MSC 2020 classification: 60L20, 60G35.
Stochastic filtering is concerned with determining the conditional distribution of an unobserved signal process \(X\) from observations of another process \(Y\), as both processes evolve continuously in time. Problems involving the filtering of stochastic systems arise frequently in numerous applications, including in finance, economics, defence and aerospace, and such problems have been studied extensively in a wide variety of settings. For a detailed account, we refer to some of the many excellent monographs on the subject, such as [1], [2], [3] and [4].
At least as far back as the late seventies, Clark [5] pointed out that it would be natural and desirable, particularly in the context of real-world applications, to obtain a robust representation of a stochastic filter, by which one typically means a continuous function of the observed path (or some features thereof), which coincides with the conditional distribution. More precisely, for a suitable class of functions \(f\), one seeks a continuous adapted function \(\Theta^f\) on the pathspace of \(Y\), such that \[\Theta^f_t(Y) = \mathbb{E}[f(X_t,Y_t) \,|\, \mathcal{F}^Y_t]\] holds almost surely, where we write \(\mathcal{F}^Y_t\) for the \(\sigma\)-algebra generated by \((Y_s)_{s \leq t}\). Such a representation is important for at least two reasons. First, it guarantees that, provided the model we suppose for the observation noise is close (in a suitable weak sense) to its true law, the resulting estimation error will be correspondingly small. In particular, it ensures that small errors in our dynamical model and in our observed data should only result in small errors in the resulting filter, and moreover that approximating the observation process by discrete-time data—as is inevitably the case in practice—leads to an accurate approximation of the conditional distribution. Second, it allows one to apply learning techniques with pointwise errors in mind (in contrast to mean squared errors) to stochastic filters, where the filter is trained as a neural network of relevant features (typically related to the topology with respect to which the filter is continuous) on training data.
In this paper, we investigate the existence of such a robust representation of the conditional distribution in a stochastic filtering model for correlated multidimensional jump-diffusion processes of the form \[\label{eq:32intro-32SDE} \begin{align} \mathrm{d}X_t &= b_1(t,X_t,Y_t) \,\mathrm{d}t + \sigma_0(t,X_t,Y_t) \,\mathrm{d}B_t + \sigma_1(t,X_t,Y_t) \,\mathrm{d}W_t\\ &\quad + \int_{\mathbb{U}_1} f_1(t,X_{t-},Y_{t-},u) \, \widetilde{N}_1(\mathrm{d}t,\mathrm{d}u) + \int_{\mathbb{U}_2} f_2(t,X_{t-},Y_{t-},u) \, \widetilde{N}_2(\mathrm{d}t,\mathrm{d}u),\\ \mathrm{d}Y_t &= b_2(t,X_t,Y_t) \,\mathrm{d}t + \sigma_2(t,Y_t) \,\mathrm{d}W_t + \int_{\mathbb{U}_2} f_3(t,Y_{t-},u) \, \widetilde{N}_2(\mathrm{d}t,\mathrm{d}u), \end{align}\tag{1}\] under essentially optimal regularity assumptions on the coefficients, where here \(B\) and \(W\) are independent Brownian motions, \(N_1\) is a Poisson random measure, and \(N_2\) is an integer-valued random measure whose compensator depends on \(X\). In particular, we consider a proper filtering problem, i.e., an equivalent change of measure exists which makes the signal disappear from the dynamics of the observation process. The case of improper filtering problems is more delicate (and generally easier).
In [5], Clark considered \(X\) and \(Y\) as diffusions driven by independent Brownian motions, and expressed the conditional distribution as a ratio of expectations via the Kallianpur–Striebel formula, under a new probability measure. The associated change of measure introduces a term involving the exponential of a stochastic integral, and finding a robust representation in the uncorrelated noise case fundamentally boils down to finding a robust version of this term. Clark suggested such a version via a formal integration by parts, and the continuity of this representation with respect to the uniform topology was also confirmed by Kushner [6] under appropriate locally uniform exponential integrability assumptions, and Clark and Crisan [7] provided a rigorous treatment of the measurability issues that arose therein. We refer to [1] for a comprehensive overview of these results.
In the correlated noise case, one needs to find a robust version of the signal itself, in terms of the observation path, in addition to the exponential terms in the Kallianpur–Striebel formula. For scalar observation, Davis used a Doss–Sussman type flow transformation to express \(X\) as the composition of a flow of a deterministic ODE driven by the sample path of the observation \(Y\), and an SDE with coefficients which are continuous in \(Y\). Paired with another formal integration by parts, this provides a robust version of the conditional distribution; for details, see [8], [9], [10] and [11]. These results were extended to a multidimensional setting by Elliott and Kohlmann [12], under some additional commutativity assumptions on the vector fields.
In a general continuous multidimensional correlated noise setting, however, it is known that there cannot exist a robust representation of the conditional distribution on the space of continuous paths endowed with the uniform topology (see [13]). Instead, Crisan, Diehl, Friz and Oberhauser [13] showed that, in such a setting, upon lifting the observation path to a rough path, there exists a representation of the filter which is continuous with respect to a suitable rough path topology.
Initiated by Lyons [14], the theory of rough paths generalizes classical notions of integration and controlled ODEs to handle highly oscillatory multidimensional paths. A rough path may be viewed as a path \(X\) which has been enhanced with its iterated integrals, which is sufficient to ensure the well-posedness and stability of solutions to nonlinear differential equations driven by such a path. In the context of stochastic analysis, such rough differential equations (RDEs) provide a robust version of the solution to the corresponding stochastic differential equation (SDE). Unlike its stochastic counterpart, such an RDE is well-posed for a fixed realisation of the driving noise, and the solution is a continuous function of the driving rough path; see, e.g., [15] or [16] for a comprehensive exposition of the theory.
Upon conditioning on the observation \(\sigma\)-algebra \(\mathcal{F}^Y_t\), it is natural to fix a lifted (in the rough path sense) realization of \((Y_s)_{s \in [0,t]}\), and consider the resulting rough dynamics. However, since the unconditioned noise also remains, the resulting equation is actually driven by both rough and stochastic noise. To make sense of such equations, in [13] the authors used a flow transformation to make sense of such mixed rough stochastic differential equations (rough SDEs), driven by a Hölder continuous rough path and a Brownian motion. They then established locally uniform exponential moments to make sense of the exponential terms appearing in the Kallianpur–Striebel formula, and hence also a locally Lipschitz continuous version of the conditional distribution.
Due to the use of flow transformations, this solution theory for rough SDEs does not provide an intrinsic solution to the equation, and comes with excessive regularity requirements on its coefficients. Over the last decade, alternative approaches to the analysis of rough SDEs have been proposed, such as the random rough path approach of, e.g., Diehl, Oberhauser and Riedel [17], in which the rough SDE is treated as an RDE driven by the random joint rough path lift of the stochastic noise together with the rough path. See also Friz and Zorin-Kranich [18] for a more recent and extensive analysis of the random rough path approach.
Another recent approach was introduced by Friz, Hocquet and Lê [19], in which the stochastic sewing lemma is used to give intrinsic meaning to rough SDEs under optimal regularity requirements on the coefficients. Since its inception, the theory of rough SDEs has received a number of theoretical extensions, such as Malliavin differentiability [20], parameter dependent rough SDEs [21], and rough SDEs with jumps [22], along with various applications to, e.g., pathwise stochastic control [23], [24], stochastic volatility models [25], and rough McKean–Vlasov equations in a filtering setting [26].
Motivated in particular by applications in mathematical finance (e.g., [27], [28]), stochastic filtering for jump-diffusions has been studied extensively, and the corresponding filtering equations have been derived in a number of settings ([29], [30], [31]). In particular, cases with correlated noise have been treated in both scalar ([32], [33], [34]) and multidimensional settings ([35], [36], [37], [38], [39], [40], [41]).
Extensions of the robustness result to filtering settings with jumps have been developed by Kushner [42] for independent noise, and by Grigelionis and Mikulevicius [38] in a multidimensional correlated noise setting under a commutativity assumption on the signal dynamics. However, a robustness result for general multidimensional correlated jump-diffusions of the form 1 , in the spirit of [13], has not yet been established. In the present paper we fill this gap, by utilizing the recent theory of rough SDEs with jumps, as developed in [22].
A key step in the procedure detailed in both [13] and the present work is to condition on one of the two noise components, in what one may refer to as a “doubly stochastic DE”, to obtain a (random) rough SDE. While it is intuitively clear that the solutions to these equations should coincide, making this rigorous requires one to establish measurability of the solution to the “randomised” rough SDE.4
Concurrently with the present work, Friz, Lê and Zhang derived a resolution to this problem in [23] and [43]. Considering solutions to rough SDEs with the driving rough path as a parameter, they establish measurability of the solution with respect to the parameter space by the application of suitable measurable selection theorems, thus obtaining the desired measurability of the randomised rough SDE. Although this approach can undoubtedly be adapted to incorporate jumps, we instead provide an alternative perspective. Rather than resorting to measurable selection, we directly adapt the classical procedure for proving well-posedness of rough SDEs to handle random rough paths, by “randomising” the metric space on which we establish an invariant contraction. This allows us to construct the solution to both the randomised rough SDE and to the corresponding doubly stochastic DE simultaneously, thus establishing that both solutions are well-defined and measurable, and that they coincide. The proof is very much in the spirit of the classical \(\alpha\)-slicing construction for the existence of solutions to SDEs driven by càdlàg semimartingales (see, e.g., [44]).
Another key ingredient we require is integrability of the exponential moments which appear in the Kallianpur–Striebel formula. The existence of exponential moments for solutions to rough SDEs has been addressed for Hölder continuous noise in [19], and a more general result of Lê [45] provides exponential moments for BMO processes via a novel version of the John–Nirenberg inequality. We extend the results of [45] to obtain bounds on so-called \(\mathrm{BMO}^{p\mathrm{-var}}\) processes, which in particular include càdlàg rough stochastic integrals.
With these ingredients in place, we establish the existence of a robust version of the conditional distribution associated with the filtering model in 1 . In general, continuity is established with respect to the rough path lift of the relevant noise components (Theorem 6), and in the case of additive noise in the observation we obtain a robustness result akin to and generalizing those in [13] and [17] (Corollary 2). Moreover, by considering the filter as a function on the space of random rough paths, we obtain a novel result on the robustness of stochastic filters with respect to model uncertainty (Theorem 7).
The paper is organised as follows. In Section 2 we discuss the necessary preliminaries from the theory of rough SDEs, as well as extensions of this theory to incorporate integrals against integer-valued random measures, and continuity with respect to Skorokhod topology. We also provide in Theorem 2 a Kolmogorov continuity-type criterion for processes with càdlàg paths. In Section 3 we establish the consistency between solutions to doubly stochastic DEs and their rough SDE counterparts, and exponential moments are then discussed in Section 4. Finally, in Section 5 we combine the ideas developed in the earlier sections to construct a robust representation of the conditional distribution in our stochastic filtering model.
Acknowledgement: The authors would like to thank Jannis Dause and Peter Friz for helpful discussions, particularly on Theorem 2 and Example 3.
For \(0 < T < \infty\), we write \(\Delta_{[0,T]} = \{(s,t) \in [0,T]^2 : s \leq t\}\). We call a function \(w \colon \Delta_{[0,T]} \to [0,\infty)\) a control on \(\Delta_{[0,T]}\) if it is superadditive, in the sense that \(w(s,u) + w(u,t) \leq w(s,t)\) for all \(s \leq u \leq t\). Note that for any control \(w\), the map \(s \mapsto w(s,t)\) is non-increasing, \(t \mapsto w(s,t)\) is non-decreasing, and \(w(t,t) = 0\) for all \(t\).
We write \(w(s,t+) := \lim_{u \searrow t} w(s,u)\) and \[w(s,t-) := \begin{cases} \lim_{u \nearrow t} w(s,u) & \text{if } \, s < t,\\ 0 & \text{if } \, s = t, \end{cases}\] and define \(w(s+,t)\) and \(w(s-,t)\) analogously.
By a partition \(\mathcal{P}\) of a given interval \([s,t]\), we mean a finite sequence of times \(\mathcal{P}= \{s = t_0 < t_1 < \cdots < t_n = t\}\). We also denote by \(|\mathcal{P}| := \max_{0 \leq i < n} |t_{i+1} - t_i|\) the mesh size of a partition \(\mathcal{P}\). We will also sometimes abuse notation slightly by writing \([u,v] \in \mathcal{P}\) to identify two consecutive times \(u, v \in \mathcal{P}\), i.e., when \(u = t_i\) and \(v = t_{i+1}\) for some \(i\).
For \(p \in [1,\infty)\), given a two-parameter function \(F \colon \Delta_{[0,T]} \to E\), taking values in any normed vector space \((E,|\cdot|)\), the \(p\)-variation of \(F\) over the interval \([s,t] \subseteq [0,T]\) is defined as \[\|F\|_{p,[s,t]} := \bigg(\sup_{\mathcal{P}\subset [s,t]} \sum_{[u,v] \in \mathcal{P}} |F_{u,v}|^p\bigg)^{\frac{1}{p}},\] where the supremum is taken over all possible partitions \(\mathcal{P}\) of the interval \([s,t]\). We say that \(F\) has finite \(p\)-variation if \(\|F\|_{p,[0,T]} < \infty\). We will sometimes also consider \(p\)-variation over general intervals, e.g., \(\|F\|_{p,[s,t)} := \lim_{u \nearrow t} \|F\|_{p,[s,u]}\).
Given a path \(A \colon [0,T] \to E\), we write \(\delta A\) for its increment function, so that \[\delta A_{s,t} = A_t - A_s\] for all \((s,t) \in \Delta_{[0,T]}\). The \(p\)-variation of \(A\) over \([s,t]\) is defined as the \(p\)-variation of its increment process, i.e., \(\|A\|_{p,[s,t]} := \|\delta A\|_{p,[s,t]}\), and we write \(V^p = V^p([0,T];E)\) for the space of (deterministic) càdlàg paths with finite \(p\)-variation.
Given a càdlàg path or process \(Y = (Y_t)_{t \in [0,T]}\), we will write \(\Delta Y_t := Y_t - Y_{t-}\) for the jump of \(Y\) at time \(t\).
For \(k \in \mathbb{N}\) and a function \(f\), we write \(\mathrm{D}^k f\) for the \(k\)-th order Fréchet derivative of \(f\). We write \(C^n_b\) for the space of functions \(f\) which are \((n-1)\)-times continuously differentiable, such that \(f\) and all its derivatives up to order \(n-1\) are bounded, and such that the \((n-1)\)-th order derivative \(\mathrm{D}^{n-1} f\) is Lipschitz continuous. We denote the corresponding norm by \(\|f\|_{C^n_b}\).
We will also write, e.g., \(\mathcal{L}(\mathbb{R}^\ell;\mathbb{R}^m)\) for the space of linear maps from \(\mathbb{R}^\ell \to \mathbb{R}^m\).
During proofs, we will often use the symbol \(\lesssim\) to indicate inequality up to a multiplicative constant. When deriving estimates, this implicit constant will depend on the same variables as the constant specified in the statement of the corresponding estimate.
We let \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t \in [0,T]},\mathbb{P})\) be a filtered probability space, and we will always assume that the filtration satisfies the usual conditions. For \(r \in [1,\infty]\), we write \(\|\cdot\|_{L^r}\) for the standard Lebesgue norm on \((\Omega,\mathcal{F},\mathbb{P})\). We also adopt the shorthand \(\mathbb{E}_s [\, \cdot \,] := \mathbb{E}[\, \cdot \,|\, \mathcal{F}_s]\) for the conditional expectation at time \(s \in [0,T]\).
As in [22], for \(q \in [1,\infty)\), \(r \in [q,\infty]\), \(s \in [0,T]\) and a random variable \(Y\), we write \[\|Y\|_{q,r,s} := \big\| \mathbb{E}_s [|Y|^q]^{\frac{1}{q}} \big\|_{L^r}\] and we write \(L^{q,r}_s\) for the space of random variables \(Y\) such that \(\|Y\|_{q,r,s} < \infty\).
For convenience, we recall the following properties of the \(\|\cdot\|_{q,r,s}\) norm, the proofs of which may be found in [22].
\((L^{q,r}_s,\|\cdot\|_{q,r,s})\) is a Banach space.
We have that \(\|Y\|_{L^q} \leq \|Y\|_{q,r,s} \leq \|Y\|_{L^r}\) and \(\| \mathbb{E}_s[Y] \|_{L^r} \leq \|Y\|_{q,r,s}\).
The norm \(\|Y\|_{q,r,s}\) is non-decreasing in each of the variables \(q, r\) and \(s\).
The following version of Hölder’s inequality, \[\label{eq:32Holder39s32inequality32for32q44r44s32norm} \big\|\mathbb{E}_s[|Y||Z|]\big\|_{L^\ell} \leq \big\|\mathbb{E}_s[|Y|^p]^{\frac{1}{p}}\big\|_{L^{p \ell}} \big\|\mathbb{E}_s[|Z|^q]^{\frac{1}{q}}\big\|_{L^{q \ell}} = \|Y\|_{p,p \ell,s} \|Z\|_{q,q \ell,s},\tag{2}\] holds for any \(\ell \in [1,\infty]\) whenever \(p, q \in (1,\infty)\) with \(\frac{1}{p} + \frac{1}{q} = 1\).
For \(p, q \in [1,\infty)\), \(r \in [q,\infty]\), and a two-parameter stochastic process \(F = (F_{s,t})_{(s,t) \in \Delta_{[0,T]}}\), we write \[\|F\|_{p,q,r,[s,t]} := \bigg(\sup_{\mathcal{P}\subset [s,t]} \sum_{[u,v] \in \mathcal{P}} \|F_{u,v}\|_{q,r,u}^p\bigg)^{\frac{1}{p}} = \bigg(\sup_{\mathcal{P}\subset [s,t]} \sum_{[u,v] \in \mathcal{P}} \big\|\mathbb{E}_u[|F_{u,v}|^q]^{\frac{1}{q}}\big\|_{L^r}^p\bigg)^{\frac{1}{p}}\] for \((s,t) \in \Delta_{[0,T]}\). For a stochastic process \(Y = (Y_t)_{t \in [0,T]}\), we then let \[\|Y\|_{p,q,r,[s,t]} := \|\delta Y\|_{p,q,r,[s,t]}.\]
We write \(V^p L^{q,r} = V^p L^{q,r}([0,T];E)\) for the space of stochastic processes \(Y \colon \Omega \times [0,T] \to E\) which are càdlàg almost surely, and are such that \(Y_0 \in L^q\) and \(\|Y\|_{p,q,r,[0,T]} < \infty\).
In the special case when \(r = q\), we also write \[\label{eq: defn p,infty,[0,T] norm} \|F\|_{p,q,[s,t]} := \|F\|_{p,q,q,[s,t]} = \bigg(\sup_{\mathcal{P}\subset [s,t]} \sum_{[u,v] \in \mathcal{P}} \|F_{u,v}\|_{L^q}^p\bigg)^{\frac{1}{p}},\tag{3}\] with \(\|Y\|_{p,q,[s,t]} := \|\delta Y\|_{p,q,[s,t]}\) and \(V^p L^q := V^p L^{q,q}\). Of course, we can also define \(\|\cdot\|_{p,\infty,[s,t]}\) and \(V^p L^\infty\), by simply replacing the \(L^q\) norm in @{eq:eq: defn p,infty,[0,T] norm} with an \(L^\infty\) norm.
We consider pairs \(\mathbf{X}= (X,\mathbb{X})\), consisting of a càdlàg path \(X \colon [0,T] \to \mathbb{R}^d\) and a càdlàg two-parameter function \(\mathbb{X}\colon \Delta_{[0,T]} \to \mathbb{R}^{d \times d}\) (where here càdlàg is understood for each time parameter separately), such that \(\|X\|_{p,[0,T]} < \infty\) and \(\|\mathbb{X}\|_{\frac{p}{2},[0,T]} < \infty\). For such pairs, and any \((s,t) \in \Delta_{[0,T]}\), we use the seminorm5 \[\|\mathbf{X}\|_{p,[s,t]} := \big( \|X\|_{p,[s,t]}^p + \|\mathbb{X}\|_{\frac{p}{2},[s,t]}^p \big)^{\frac{1}{p}},\] which induces the pseudometric \[(\mathbf{X},\widetilde{\mathbf{X}}) \, \mapsto \, \|\mathbf{X}- \widetilde{\mathbf{X}}\|_{p,[s,t]} = \big( \|X - \widetilde{X}\|_{p,[s,t]}^p + \|\mathbb{X}- \widetilde{\mathbb{X}}\|_{\frac{p}{2},[s,t]}^p \big)^{\frac{1}{p}}\] for pairs \(\mathbf{X}= (X,\mathbb{X})\) and \(\widetilde{\mathbf{X}}= (\widetilde{X},\widetilde{\mathbb{X}})\).
As defined in [46], for a given \(p \in [2,3)\), a càdlàg rough path is such a pair \(\mathbf{X}= (X,\mathbb{X})\), such that \(\|\mathbf{X}\|_{p,[0,T]} < \infty\), and such that Chen’s relation \(\mathbb{X}_{s,t} = \mathbb{X}_{s,u} + \mathbb{X}_{u,t} + \delta X_{s,u} \otimes \delta X_{u,t}\) holds for all \(0 \leq s \leq u \leq t \leq T\). We write \(\mathscr{V}^p = \mathscr{V}^p([0,T];\mathbb{R}^d)\) for the space of càdlàg rough paths.
We will sometimes write \(\Delta \mathbb{X}_t := \mathbb{X}_{t-,t}\) for the “jump” of \(\mathbb{X}\) at time \(t\), and we will also use the shorthand \(|\Delta \mathbf{X}_t| := |\Delta X_t| + |\Delta \mathbb{X}_t|\).
Given a two-parameter process \(A = (A_{s,t})_{(s,t) \in \Delta_{[0,T]}}\), we write \(\mathbb{E}_{\boldsymbol{\cdot}} A\) for the two-parameter process given by \[(\mathbb{E}_{\boldsymbol{\cdot}} A)_{s,t} := \mathbb{E}_s [A_{s,t}]\] for every \((s,t) \in \Delta_{[0,T]}\).
Definition 1 (Definition 4.1 in [22]). Let \(p \in [2,3)\), \(q \in [2,\infty)\) and \(r \in [q,\infty]\), and let \(X \in V^p\). We call a pair of processes \((Y,Y')\) a stochastic controlled path (relative to \(X\)), if \(Y\) and \(Y'\) are both adapted, \(Y \in V^p L^{q,r}\), \(Y' \in V^p L^{q,r}\), \(\sup_{s \in [0,T]} \|Y'_s\|_{L^r} < \infty\), and \(\|\mathbb{E}_{\boldsymbol{\cdot}} R^Y\|_{\frac{p}{2},r,[0,T]} < \infty\), where the two-parameter process \(R^Y = (R^Y_{s,t})_{(s,t) \in \Delta_{[0,T]}}\) is defined by \[\delta Y_{s,t} = Y'_s \delta X_{s,t} + R^Y_{s,t}\] for every \((s,t) \in \Delta_{[0,T]}\).
We write \(\mathcal{V}^{p,q,r}_X\) for the space of stochastic controlled paths relative to \(X\).
Given a rough path \(\mathbf{X}\in \mathscr{V}^p\) and a stochastic controlled path \((Y,Y') \in \mathcal{V}^{p,q,r}_X\), one can consider the rough stochastic integral of \((Y,Y')\) with respect to \(\mathbf{X}\), as defined in the following lemma.
Lemma 1 (Lemmas 4.3 and 4.6 in [22]). Let \(p \in [2,3)\), \(q \in [2,\infty)\) and \(r \in [q,\infty]\). Let \(\mathbf{X}= (X,\mathbb{X}) \in \mathscr{V}^p\) be a càdlàg rough path, and let \((Y,Y') \in \mathcal{V}^{p,q,r}_X\) be a stochastic controlled path. Then there exists an \(L^q\)-integrable adapted càdlàg process \(\int_0^\cdot Y_u \,\mathrm{d}\mathbf{X}_u\), such that, for every \((s,t) \in \Delta_{[0,T]}\), \[\lim_{|\mathcal{P}| \to 0} \bigg\|\int_s^t Y_u \,\mathrm{d}\mathbf{X}_u - \sum_{[u,v] \in \mathcal{P}} \big(Y_u \delta X_{u,v} + Y'_u \mathbb{X}_{u,v}\big)\bigg\|_{q,r,s} = 0,\] where the limit holds along partitions \(\mathcal{P}\) of the interval \([s,t]\) as the mesh size tends to zero.
In our application to stochastic filtering in Section 5, it will be convenient to express the stochastic noise generated by a pure jump process in the language of random measures. We refer to, e.g., [44] or [47] for a detailed exposition on random measures and their associated stochastic integration. For convenience, in this section we will begin by fixing some standard notation, and recalling a few fundamental properties.
We will write \((\mathbb{U},\mathcal{U})\) to denote a given Blackwell space, and consider an integer-valued random measure \(N\) on \([0,T] \times \mathbb{U}\). Let us write \(\nu\) for the predictable compensator of \(N\). We recall that there exists a predictable, non-decreasing and integrable process \(A\), and a kernel \(K\), such that \[\label{eq:32decomposition32of32nu} \nu(\mathrm{d}t,\mathrm{d}u) = K(t,\mathrm{d}u) \,\mathrm{d}A_t.\tag{4}\] In general the process \(A\) here is càdlàg, but, for our purposes, it will always be assumed to be continuous. In particular, this means that, almost surely, \(\nu(\{t\} \times \mathbb{U}) = 0\) for each \(t \in [0,T]\).
If \(N\) is a Poisson measure, then its compensator \(\nu\) is deterministic, and if \(N\) is a homogeneous Poisson measure, then \(\nu(\mathrm{d}t,\mathrm{d}u) = F(\mathrm{d}u) \,\mathrm{d}t\) for some (deterministic) \(\sigma\)-finite measure \(F\).
Lemma 2 (Ch. II, Proposition 1.14 in [47]). Let \(N\) be an integer-valued random measure on \([0,T] \times \mathbb{U}\). Then there exists an optional \(\mathbb{U}\)-valued process \(\beta\), and a thin random set \(D\), such that, for almost every \(\omega \in \Omega\), \[\label{eq:32definition32D95i32and32beta94i} N(\omega;\mathrm{d}t,\mathrm{d}u) = \sum_{0 < s \leq T} \mathbf{1}_D(\omega,s) \delta_{(s,\beta_s(\omega))}(\mathrm{d}t,\mathrm{d}u),\tag{5}\] where \(\delta_{(s,u)}\) denotes the Dirac measure at the point \((s,u)\).
In this setting (in particular with \(A\) assumed to be continuous), given an integer-valued random measure \(N\), we denote by \(G_{\mathrm{loc}}(N)\) the set of predictable functions \(\zeta\) on \(\Omega \times [0,T] \times \mathbb{U}\) such that the process \((\sum_{s \leq \cdot} |\zeta(\omega;s,\beta_s(\omega))|^2 \mathbf{1}_D(\omega,s))^{\frac{1}{2}}\) is locally integrable. Given \(\zeta \in G_{\mathrm{loc}}(N)\), we call the stochastic integral of \(\zeta\) with respect to \(\widetilde{N} := N - \nu\), denoted \(\int_0^\cdot \int_{\mathbb{U}} \zeta(s,u) \, \widetilde{N}(\mathrm{d}s,\mathrm{d}u)\), the purely discontinuous local martingale whose jumps coincide with the process \(\zeta(\cdot,\beta) \mathbf{1}_D\).
Assumption 1. Let \(\nu\) be a predictable random measure on \([0,T] \times \mathbb{U}\), and suppose that \(A\) and \(K\) are a process and kernel such that 4 holds. We assume that the process \(A\) is almost surely continuous, and that \(A \in V^{\frac{p}{2}} L^{\frac{q}{2},\infty} \cap V^{\frac{p}{q}} L^{1,\infty}\), for some \(2 \leq q \leq p < 3\). Let \(g\) be a measurable function on \([0,T] \times \mathbb{R}^m \times \mathbb{U}\). We assume that \[\bigg\| \sup_{s \in [0,T]} \int_{\mathbb{U}} \big( |g(s,0,u)|^2 \vee |g(s,0,u)|^q \big) \, K(s,\mathrm{d}u) \bigg\|_{L^\infty} < \infty,\] and, additionally, that there exists a constant \(C > 0\) such that, for each \(\ell \in \{2,q\}\), any \(y, \widetilde{y}\in \mathbb{R}^m\) and any \(s \in [0,T]\), \[\int_{\mathbb{U}} |g(s,y,u) - g(s,\widetilde{y},u)|^\ell \, K(s,\mathrm{d}u) \leq C (1 \wedge |y - \widetilde{y}|^\ell)\] holds \(\mathbb{P}\)-almost surely.
In particular, if \(N\) is an integer-valued random measure with compensator \(\nu\), and if \(g\) is a measurable function on \([0,T] \times \mathbb{R}^m \times \mathbb{U}\), then, under Assumption 1, it is straightforward to see that \((g(s,Z_s,\cdot))_{s \in [0,T]} \in G_{\mathrm{loc}}(N)\) for any \(\mathbb{R}^m\)-valued predictable process \(Z\).
We consider the rough SDE given by \[\label{eq:32RSDE32with32measure} Y_t = y_0 + \int_0^t b(Y_s) \,\mathrm{d}s + \int_0^t \sigma(Y_{s-}) \,\mathrm{d}M_s + \int_0^t \int_{\mathbb{U}} g(s,Y_{s-},u) \, \widetilde{N}(\mathrm{d}s,\mathrm{d}u) + \int_0^t f(Y_s) \,\mathrm{d}\mathbf{X}_s\tag{6}\] for \(t \in [0,T]\), where \(M\) is a càdlàg martingale, \(\widetilde{N} = N - \nu\) is a compensated integer-valued random measure, and \(\mathbf{X}\) is a càdlàg rough path. In particular, the second integral is an Itô integral against \(M\), the third integral is a stochastic integral against \(\widetilde{N}\) in the sense of random measures, and the last integral is the rough stochastic integral of \((f(Y),\mathrm{D}f(Y) Y')\) against \(\mathbf{X}\), in the sense of Lemma 1.
The main result of [22] was to establish existence, uniqueness and stability of solutions to rough SDEs driven by a càdlàg martingale and a càdlàg rough path. It is straightforward to extend this result to include an integer-valued random measure, which we state precisely in the theorem below. Since the proof is just a straightforward adaptation of the proof of [22], using the estimate in part (ii) of Lemma 16, we omit the proof here for brevity.
Theorem 1. Let \(2 \leq q \leq p < 3\), \(b \in C^1_b\), \(\sigma \in C^1_b\), \(f \in C^3_b\) and \(g \colon [0,T] \times \mathbb{R}^m \times \mathbb{U}\to \mathbb{R}^m\). Let \(y_0 \in L^q\) be \(\mathcal{F}_0\)-measurable, \(\mathbf{X}= (X,\mathbb{X}) \in \mathscr{V}^p\) be a càdlàg rough path, and let \(M \in V^p L^{q,\infty}\) be a càdlàg martingale. Further, let \(N\) be an integer-valued random measure on \([0,T] \times \mathbb{U}\) for some Blackwell space \((\mathbb{U},\mathcal{U})\), with compensator \(\nu\), write \(\widetilde{N}= N - \nu\) for the corresponding compensated random measure, and suppose that \(\nu\) and \(g\) satisfy Assumption 1.
Then there exists a process \(Y\), which is unique up to indistinguishability, such that \(Y\) has almost surely càdlàg sample paths, \((Y,Y') \in \mathcal{V}^{p,q,\infty}_X\) is a stochastic controlled path, where \(Y' = f(Y)\), and such that, almost surely, 6 holds for every \(t \in [0,T]\).
Moreover, if \(y_0, \widetilde{y}_0 \in L^q\) are \(\mathcal{F}_0\)-measurable, \(\mathbf{X}, \widetilde{\mathbf{X}}\in \mathscr{V}^p\) are two càdlàg rough paths, and \(M, \widetilde{M}\in V^p L^{q,\infty}\) are càdlàg martingales, such that the norms \(\|\mathbf{X}\|_{p,[0,T]}\), \(\|\widetilde{\mathbf{X}}\|_{p,[0,T]}\), \(\|M\|_{p,q,\infty,[0,T]}\) and \(\|\widetilde{M}\|_{p,q,\infty,[0,T]}\) are all bounded by some constant \(L > 0\), and if \(Y, \widetilde{Y}\) are the solutions of 6 corresponding to the data \((y_0,M,N,\mathbf{X})\) and \((\widetilde{y}_0,\widetilde{M},N,\widetilde{\mathbf{X}})\) respectively, then we have that \[\label{eq:32Lipschitz32continuity32of32solution32map} \begin{align} \|Y &- \widetilde{Y}\|_{p,q,[0,T]} + \|Y' - \widetilde{Y}'\|_{p,q,[0,T]} + \|\mathbb{E}_{\boldsymbol{\cdot}} (R^Y - R^{\widetilde{Y}})\|_{\frac{p}{2},q,[0,T]}\\ &\leq C \big(\|y_0 - \widetilde{y}_0\|_{L^q} + \|M - \widetilde{M}\|_{p,q,[0,T]} + \|\mathbf{X}- \widetilde{\mathbf{X}}\|_{p,[0,T]}\big), \end{align}\tag{7}\] where the constant \(C\) depends only on \(p, q, \|b\|_{C^1_b}, \|\sigma\|_{C^1_b}, \|f\|_{C^3_b}, g, \nu, T\) and \(L\).
We denote by \(\Lambda\) the set of increasing bijective functions from \([0,T] \to [0,T]\). Given a \(\lambda \in \Lambda\) and a rough path \(\mathbf{X}= (X,\mathbb{X}) \in \mathscr{V}^p\), we write \(\mathbf{X}\circ \lambda := (X \circ \lambda, \mathbb{X}\circ (\lambda,\lambda))\) for the rough path obtained from \(\mathbf{X}\) by the “time change” \(\lambda\).
As in [46], given rough paths \(\mathbf{X}, \mathbf{Z}\in \mathscr{V}^p\), we define the \(p\)-variation J1-Skorokhod distance by \[\label{eq:32defn32skorokhod32metric} \sigma_{p,[0,T]}(\mathbf{X},\mathbf{Z}) := \inf_{\lambda \in \Lambda} \big\{ |\lambda| \vee \|\mathbf{X}\circ \lambda - \mathbf{Z}\|_{p,[0,T]} \big\},\tag{8}\] where \(|\lambda| := \sup_{t \in [0,T]} |\lambda(t) - t|\).
Proposition 2. Let \(2 \leq q \leq p < 3\), \(b \in C^1_b\), \(\sigma \in C^1_b\), \(f \in C^3_b\) and \(g \colon [0,T] \times \mathbb{R}^m \times \mathbb{U}\to \mathbb{R}^m\). Let \(\mathbf{X}\) and \((\mathbf{X}^n)_{n \in \mathbb{N}}\) be càdlàg rough paths, let \(y_0 \in L^q\) be \(\mathcal{F}_0\)-measurable, let \(N\) be an integer-valued random measure with compensator \(\nu\), such that \(\nu\) and \(g\) satisfy Assumption 1, and let \(M \in V^p L^{q,\infty}\) be a càdlàg martingale which is continuous at deterministic times.
Let us write \(Y\) and \(Y^n\) for the solutions to the rough SDE 6 with data \((y_0,M,N,\mathbf{X})\) and \((y_0,M,N,\mathbf{X}^n)\) respectively. If \(\sigma_{p,[0,T]}(\mathbf{X}^n,\mathbf{X}) \to 0\) as \(n \to \infty\), then \(Y^n_T \to Y_T\) in \(L^q\) as \(n \to \infty\).
Remark 3. We note that the assumption in Proposition 2 that \(M\) is continuous at deterministic times, and the assumption that the process \(A\) in 4 is continuous, are both necessary. Indeed, the result of Proposition 2 is not true in general if either \(M\) or \(N\) has a positive probability of having a jump at any deterministic time \(t \in (0,T]\). Essentially, in this case, time-changing the driving rough path \(\mathbf{X}\) can change the order of events, which fundamentally alters the solution, violating the desired continuity, as illustrated in the following example. Of course, this assumption on \(M\) is satisfied, for instance, by any centred Lévy process.
Example 1. Let \(\xi\) be an \(\mathcal{F}_{t_0}\)-measurable integrable random variable for some \(t_0 \in (0,T)\), such that \(\mathbb{E}[\xi \,|\, \mathcal{F}_{t_0-}] = 0\), so that the process \(M = (M_t)_{t \in [0,T]}\) given by \(M_t := \xi \mathbf{1}_{[t_0,T]}(t)\) is a martingale. Further, let \(X_t = \mathbf{1}_{[t_0,T]}(t)\) and \(X^n_t = \mathbf{1}_{[t_0 - \frac{1}{n},T]}(t)\) so that, in particular, \(X^n \to X\) in Skorokhod topology. Let \(Y^n\) be the solution to the SDE \[\mathrm{d}Y^n_t = Y^n_{t-} \,\mathrm{d}M_t + \mathrm{d}X^n_t,\] and let \(Y\) be the solution to the same SDE with \(X^n\) replaced by \(X\). One can then directly check that \(Y_T = 1\), while \(Y^n_T = 1 + \xi\) for every \(n \in \mathbb{N}\), so that \(Y^n_T\) does not converge to \(Y_T\). This example illustrates that, if \(M\) (or \(N\)) has a jump at a deterministic time, then we cannot expect the solution to a rough SDE to be continuous in the sense of Proposition 2.
Proof of Proposition 2. We first note that, since \(\Delta_{[0,T]}\) is compact, it follows from our assumptions that the function \(\Delta_{[0,T]} \ni (s,t) \mapsto \|M\|_{p,q,[s,t]} + \|A\|_{\frac{p}{2},\frac{q}{2},[s,t]} + \|A\|_{\frac{p}{q},1,[s,t]}\) is uniformly continuous, so that, in particular, \[\label{eq:32condition32on32martingale32for32Skorokhod32continuity} \lim_{h \searrow 0} \, \sup_{|t - s| < h} \big(\|M\|_{p,q,[s,t]} + \|A\|_{\frac{p}{2},\frac{q}{2},[s,t]} + \|A\|_{\frac{p}{q},1,[s,t]}\big) = 0.\tag{9}\]
Let \(C > 0\) be the Lipschitz constant in 7 , which may be chosen such that 7 holds for \(\mathbf{X}\) and \(\mathbf{X}^n\) for every \(n \in \mathbb{N}\).
Since \(\sigma_{p,[0,T]}(\mathbf{X}^n,\mathbf{X}) \to 0\) as \(n \to \infty\), there exists a sequence \((\lambda_n)_{n \in \mathbb{N}} \subset \Lambda\) such that \[|\lambda_n| \to 0 \qquad \text{and} \qquad \|\mathbf{X}^n - \mathbf{X}\circ \lambda_n\|_{p,[0,T]} \to 0 \quad \text{as} \quad n \to \infty.\]
Let \(\varepsilon> 0\), and let \(Y^{(\lambda_n)}\) be the solution to the rough SDE 6 with data \((y_0,M,N,\mathbf{X}\circ \lambda_n)\). Then \[\label{eq:32Y94n95T32-32Y94lambda95T32est} \|Y^n_T - Y^{(\lambda_n)}_T\|_{L^q} \leq \|Y^n - Y^{(\lambda_n)}\|_{p,q,[0,T]} \leq C \|\mathbf{X}^n - \mathbf{X}\circ \lambda_n\|_{p,[0,T]} < \varepsilon\tag{10}\] for all sufficiently large \(n\).
Given a partition \(\mathcal{P}= \{0 = t_0 < t_1 < \cdots < t_N = T\}\) of the interval \([0,T]\), we write \(\mathbf{X}^\mathcal{P}\) for the piecewise constant approximation of \(\mathbf{X}\) along \(\mathcal{P}\). That is, \(\mathbf{X}^\mathcal{P}= (X^\mathcal{P},\mathbb{X}^\mathcal{P})\), where \(X^\mathcal{P}_u = X_{t_i}\) and \(\mathbb{X}^\mathcal{P}_{u,v} = \mathbb{X}_{t_i,t_j}\) whenever \(u \in [t_i,t_{i+1})\) and \(v \in [t_j,t_{j+1})\).
It is straightforward to see6 that, for any \(p' \in (p,3)\), there exists a partition \(\mathcal{P}= \{0 = t_0 < t_1 < \cdots < t_N = T\}\) of the interval \([0,T]\) such that \[\|\mathbf{X}^\mathcal{P}- \mathbf{X}\|_{p',[0,T]} < \frac{\varepsilon}{C}.\] Since \(p\)-variation is invariant under time changes, we have that \(\|\mathbf{X}^\mathcal{P}\circ \lambda_n - \mathbf{X}\circ \lambda_n\|_{p',[0,T]} < \frac{\varepsilon}{C}\) also holds for each \(n \in \mathbb{N}\). Writing \(Y^\mathcal{P}\) and \(Y^{\lambda_n^{-1}(\mathcal{P})}\) for the solutions to the rough SDE with data \((y_0,M,N,\mathbf{X}^\mathcal{P})\) and \((y_0,M,N,\mathbf{X}^{\mathcal{P}} \circ \lambda_n)\) respectively, we then have that \[\label{eq:32Y94cP95T32-32Y95T32estimate} \|Y^\mathcal{P}_T - Y_T\|_{L^q} \leq \|Y^\mathcal{P}- Y\|_{p',q,[0,T]} \leq C \|\mathbf{X}^\mathcal{P}- \mathbf{X}\|_{p',[0,T]} < \varepsilon\tag{11}\] and, similarly, \[\label{eq:32Y94lambdacP95T32-32Y94lambda95T32estimate} \|Y^{\lambda_n^{-1}(\mathcal{P})}_T - Y^{(\lambda_n)}_T\|_{L^q} \leq \|Y^{\lambda_n^{-1}(\mathcal{P})} - Y^{(\lambda_n)}\|_{p',q,[0,T]} \leq C \|\mathbf{X}^{\mathcal{P}} \circ \lambda_n - \mathbf{X}\circ \lambda_n\|_{p',[0,T]} < \varepsilon.\tag{12}\]
We now take \(n \in \mathbb{N}\) sufficiently large such that \(|\lambda_n| < \frac{1}{2} \min_{0 \leq i < N} |t_{i+1} - t_i|\). This means in particular that \(t_{i-1} \vee \lambda_n^{-1}(t_{i-1}) < t_i \wedge \lambda_n^{-1}(t_i)\) for each \(i\). Let us consider the solutions \(Y^\mathcal{P}\) and \(Y^{\lambda_n^{-1}(\mathcal{P})}\) of the rough SDE driven by the piecewise constant rough paths \(\mathbf{X}^\mathcal{P}\) and \(\mathbf{X}^\mathcal{P}\circ \lambda_n\) respectively. In particular, \(\mathbf{X}^\mathcal{P}\) only jumps at the times \(t_i\) for \(i = 1, \ldots, N\), and \(\mathbf{X}^\mathcal{P}\circ \lambda_n\) only jumps at the times \(\lambda_n^{-1}(t_i)\) for \(i = 1, \ldots, N\), and the jump sizes are given by \[\Delta X^\mathcal{P}_{t_i} = \Delta (X^\mathcal{P}\circ \lambda_n)_{\lambda_n^{-1}(t_i)} = \delta X_{t_{i-1},t_i} \quad \text{and} \quad \Delta \mathbb{X}^{\mathcal{P}}_{t_i} = \Delta (\mathbb{X}^{\mathcal{P}} \circ (\lambda_n,\lambda_n))_{\lambda_n^{-1}(t_i)} = \mathbb{X}_{t_{i-1},t_i}.\]
For some \(i\), let us suppose that \(t_i < \lambda_n^{-1}(t_i)\). We then have that \[\begin{align} Y^\mathcal{P}_{\lambda_n^{-1}(t_i)} &= Y^\mathcal{P}_{t_i-} + f(Y^\mathcal{P}_{t_i-}) \delta X_{t_{i-1},t_i} + \mathrm{D}f(Y^\mathcal{P}_{t_i-}) f(Y^\mathcal{P}_{t_i-}) \mathbb{X}_{t_{i-1},t_i}\\ &\quad + \int_{t_i}^{\lambda_n^{-1}(t_i)} b(Y^\mathcal{P}_s) \,\mathrm{d}s + \int_{t_i}^{\lambda_n^{-1}(t_i)} \sigma(Y^\mathcal{P}_s) \,\mathrm{d}M_s + \int_{t_i}^{\lambda^{-1}_n(t_i)} \int_{\mathbb{U}} g(s,Y^\mathcal{P}_s,u) \, \widetilde{N}(\mathrm{d}s,\mathrm{d}u) \end{align}\] and \[\begin{align} &Y^{\lambda_n^{-1}(\mathcal{P})}_{\lambda_n^{-1}(t_i)} = Y^{\lambda_n^{-1}(\mathcal{P})}_{t_i-} + f\big(Y^{\lambda_n^{-1}(\mathcal{P})}_{\lambda_n^{-1}(t_i)-}\big) \delta X_{t_{i-1},t_i} + \mathrm{D}f\big(Y^{\lambda_n^{-1}(\mathcal{P})}_{\lambda_n^{-1}(t_i)-}\big) f\big(Y^{\lambda_n^{-1}(\mathcal{P})}_{\lambda_n^{-1}(t_i)-}\big) \mathbb{X}_{t_{i-1},t_i}\\ &+ \int_{t_i}^{\lambda_n^{-1}(t_i)} b(Y^{\lambda_n^{-1}(\mathcal{P})}_s) \,\mathrm{d}s + \int_{t_i}^{\lambda_n^{-1}(t_i)} \sigma(Y^{\lambda_n^{-1}(\mathcal{P})}_s) \,\mathrm{d}M_s + \int_{t_i}^{\lambda^{-1}_n(t_i)} \int_{\mathbb{U}} g(s,Y^{\lambda_n^{-1}(\mathcal{P})}_s,u) \, \widetilde{N}(\mathrm{d}s,\mathrm{d}u). \end{align}\]
By the BDG inequality and [22], it is straightforward to see that \[\bigg\|\int_{t_i}^{\lambda_n^{-1}(t_i)} b(Y^\mathcal{P}_s) \,\mathrm{d}s + \int_{t_i}^{\lambda_n^{-1}(t_i)} \sigma(Y^\mathcal{P}_s) \,\mathrm{d}M_s\bigg\|_{L^q} \lesssim (\lambda_n^{-1}(t_i) - t_i) + \|M\|_{p,q,[t_i,\lambda_n^{-1}(t_i)]},\] and similarly, applying part (ii) of Lemma 16 with \(r = q\), that \[\bigg\|\int_{t_i}^{\lambda^{-1}_n(t_i)} \int_{\mathbb{U}} g(s,Y^\mathcal{P}_s,u) \, \widetilde{N}(\mathrm{d}s,\mathrm{d}u)\bigg\|_{L^q} \lesssim \|A\|_{\frac{p}{2},\frac{q}{2},[t_i,\lambda^{-1}_n(t_i)]}^{\frac{1}{2}} + \|A\|_{\frac{p}{q},1,[t_i,\lambda^{-1}_n(t_i)]}^{\frac{1}{q}}.\] Clearly, the same holds true with \(Y^\mathcal{P}\) replaced by \(Y^{\lambda_n^{-1}(\mathcal{P})}\), and, by the condition in 9 , the right-hand side above can be made arbitrarily small by taking \(n\) sufficiently large.
We also have that \[\begin{align} &\big\|f(Y^\mathcal{P}_{t_i-}) \delta X_{t_{i-1},t_i} + \mathrm{D}f(Y^\mathcal{P}_{t_i-}) f(Y^\mathcal{P}_{t_i-}) \mathbb{X}_{t_{i-1},t_i}\\ &\quad - f\big(Y^{\lambda_n^{-1}(\mathcal{P})}_{\lambda_n^{-1}(t_i)-}\big) \delta X_{t_{i-1},t_i} - \mathrm{D}f\big(Y^{\lambda_n^{-1}(\mathcal{P})}_{\lambda_n^{-1}(t_i)-}\big) f\big(Y^{\lambda_n^{-1}(\mathcal{P})}_{\lambda_n^{-1}(t_i)-}\big) \mathbb{X}_{t_{i-1},t_i}\big\|_{L^q}\\ &\lesssim \big\|Y^\mathcal{P}_{t_i-} - Y^{\lambda_n^{-1}(\mathcal{P})}_{\lambda_n^{-1}(t_i)-}\big\|_{L^q} \big(|\delta X_{t_{i-1},t_i}| + |\mathbb{X}_{t_{i-1},t_i}|\big)\\ &\leq \bigg\|Y^\mathcal{P}_{t_i-} - Y^{\lambda_n^{-1}(\mathcal{P})}_{t_i-} - \int_{t_i}^{\lambda_n^{-1}(t_i)} b(Y^{\lambda_n^{-1}(\mathcal{P})}_s) \,\mathrm{d}s\\ &\qquad - \int_{t_i}^{\lambda_n^{-1}(t_i)} \sigma(Y^{\lambda_n^{-1}(\mathcal{P})}_s) \,\mathrm{d}M_s - \int_{t_i}^{\lambda^{-1}_n(t_i)} \int_{\mathbb{U}} g(s,Y^{\mathcal{P}}_s,u) \, \widetilde{N}(\mathrm{d}s,\mathrm{d}u)\bigg\|_{L^q} \|\mathbf{X}\|_{p,[0,T]}. \end{align}\] It follows that \[\big\|Y^\mathcal{P}_{\lambda_n^{-1}(t_i)} - Y^{\lambda_n^{-1}(\mathcal{P})}_{\lambda_n^{-1}(t_i)}\big\|_{L^q} \leq c \big\|Y^\mathcal{P}_{t_i-} - Y^{\lambda_n^{-1}(\mathcal{P})}_{t_i-}\big\|_{L^q} + \eta,\] where \(c > 0\) is a constant which depends only on \(\|f\|_{C^3_b}\) and \(\|\mathbf{X}\|_{p,[0,T]}\), and \(\eta > 0\) is a constant which can be chosen arbitrarily small by taking \(n\) sufficiently large. The cases when \(t_i > \lambda_n^{-1}(t_i)\) and when \(t_i = \lambda_n^{-1}(t_i)\) may be treated similarly, and hence, in every case, we have that \[\label{eq:32bound32with32X32and32eta} \big\|Y^\mathcal{P}_{t_i \vee \lambda_n^{-1}(t_i)} - Y^{\lambda_n^{-1}(\mathcal{P})}_{t_i \vee \lambda_n^{-1}(t_i)}\big\|_{L^q} \leq c \big\|Y^\mathcal{P}_{(t_i \wedge \lambda_n^{-1}(t_i))-} - Y^{\lambda_n^{-1}(\mathcal{P})}_{(t_i \wedge \lambda_n^{-1}(t_i))-}\big\|_{L^q} + \eta.\tag{13}\] Since \(\mathbf{X}^\mathcal{P}\) and \(\mathbf{X}^\mathcal{P}\circ \lambda_n\) are constant on the interval \([t_{i-1} \vee \lambda_n^{-1}(t_{i-1}),t_i \wedge \lambda_n^{-1}(t_i))\) for each \(i\), by again using the stability of solutions to rough SDEs, we can simply estimate \[\label{eq:32bound32with32C} \big\|Y^\mathcal{P}_{(t_i \wedge \lambda_n^{-1}(t_i))-} - Y^{\lambda_n^{-1}(\mathcal{P})}_{(t_i \wedge \lambda_n^{-1}(t_i))-}\big\|_{L^q} \leq C \big\|Y^\mathcal{P}_{t_{i-1} \vee \lambda_n^{-1}(t_{i-1})} - Y^{\lambda_n^{-1}(\mathcal{P})}_{t_{i-1} \vee \lambda_n^{-1}(t_{i-1})}\big\|_{L^q}.\tag{14}\]
Since \(Y^\mathcal{P}_0 - Y^{\lambda_n^{-1}(\mathcal{P})}_0 = 0\), by combining the estimates in 13 and 14 for each \(i = 1, \ldots, N\), we deduce that \(\|Y^\mathcal{P}_T - Y^{\lambda_n^{-1}(\mathcal{P})}_T\|_{L^q} \lesssim \eta\), where the implicit multiplicative constant depends only on \(C\), \(\|f\|_{C^3_b}\) and \(\|\mathbf{X}\|_{p,[0,T]}\). Thus, by choosing \(n\) sufficiently large, we can ensure that \(\eta\) is sufficiently small to guarantee that \(\|Y^\mathcal{P}_T - Y^{\lambda_n^{-1}(\mathcal{P})}_T\|_{L^q} < \varepsilon\). By combining this with 10 , 11 and 12 , we deduce that \(\|Y^n_T - Y_T\|_{L^q} < 4 \varepsilon\) for all sufficiently large \(n\), so that \(Y^n_T \to Y_T\) in \(L^q\) as \(n \to \infty\). ◻
Remark 4. Using the same approach, the proof of Proposition 2 can be extended slightly to also obtain the convergence of rough stochastic integrals with respect to the Skorokhod topology. More precisely, under the assumptions of Proposition 2, for any \(h \in C^2_b\), if \(\sigma_{p,[0,T]}(\mathbf{X}^n,\mathbf{X}) \to 0\) as \(n \to \infty\), then \(\int_0^T h(Y^n_s) \,\mathrm{d}\mathbf{X}^n_s \to \int_0^T h(Y_s) \,\mathrm{d}\mathbf{X}_s\) in \(L^q\) as \(n \to \infty\).
We conclude this section with a variant of the Kolmogorov continuity criterion adapted to the setting of discontinuous paths with finite \(p\)-variation. Such a result appears to be unexplored in the rough path literature, but the authors are aware of an analogous result in a discrete-time setting in [48]. It is claimed in [49] that a similar result holds in continuous-time, but we are not aware of any publicly available sources for such a result. The corresponding result for continuous paths7 is already contained in corresponding Besov-space embeddings; see, e.g., [51] or [50]. Using these embeddings, the authors of [52] establish a Kolmogorov-type criterion for piecewise constant random rough paths with finitely many deterministic jumps. Our result allows one to extend this result to the general càdlàg setting, without requiring either finitely many deterministic jumps or a loss of regularity depending on the integrability assumptions.
We emphasize that our proof is much more constructive and akin to classical treatments of \(p\)-variation than the ones in the previously mentioned works. In particular, we do not resort to interpolation spaces or duality-type arguments, but instead work directly at the level of paths and controls. One may also interpret the result as a form of Minkowski’s integral inequality for \(p\)-variation paths in Lebesgue spaces.
Theorem 2. Let \(1 \leq p < q < \infty\), and let \(Y\) be a stochastic process which is right-continuous in probability and such that \(\|Y\|_{p,q,[0,T]} < \infty\). Then there exists a modification of \(Y\) with sample paths which are almost surely càdlàg and such that, for every \(\tilde{p}> p\), we have that \[\label{eq:32Kolmogorov32estimate} \big\| \|Y\|_{\tilde{p},[0,T]} \big\|_{L^q} \leq C \|Y\|_{p,q,[0,T]},\tag{15}\] where the constant \(C\) depends only on \(p\) and \(\tilde{p}\).
Conversely, if \(1 \leq q \leq p < \infty\), and if \(Y\) is any process with almost surely càdlàg sample paths, then \[\|Y\|_{p,q,[0,T]} \leq \big\| \|Y\|_{p,[0,T]} \big\|_{L^q}.\]
The proof of Theorem 2 is given in Appendix 6.4. We note that both of the conditions \(p < q\) and \(\tilde{p}> p\) in 15 are sharp. Indeed, in Example 3 we provide a counterexample for when either of these conditions is not satisfied.
Throughout this section, we let \((\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \in [0,T]}, \mathbb{P})\) and \((\bar{\Omega}, \bar{\mathcal{F}}, (\bar{\mathcal{F}}_t)_{t \in [0,T]}, \bar{\mathbb{P}})\) be two filtered probability spaces, and denote by \[(\hat{\Omega}, \hat{\mathcal{F}}, (\hat{\mathcal{F}}_t)_{t \in [0,T]}, \hat{\mathbb{P}}) := (\Omega \times \bar{\Omega}, \mathcal{F}\otimes \bar{\mathcal{F}}, (\mathcal{F}_t \otimes \bar{\mathcal{F}}_t)_{t \in [0,T]}, \mathbb{P}\otimes \bar{\mathbb{P}})\] the corresponding filtered product probability space.8 In particular, \(\hat{\Omega}\) indicates that we are considering a product of two spaces (bringing them under one roof). In contrast to many probabilistic results, we actually do need this inner structure of the probability space, since on the one hand we are dealing with Itô rough path lifts of semimartingales (whose randomness lives on \(\Omega\)), and on the other hand we have additional randomness (which lives on \(\bar{\Omega}\)), and these need to be separated and handled with care.
In the following, we will write, e.g., \(\|\cdot\|_{q,r,s,\bar{\Omega}}\) for the norm \(\|\cdot\|_{q,r,s}\) when we wish to emphasize that the norm is taken with respect to the probability space \((\bar{\Omega}, \bar{\mathcal{F}}, (\bar{\mathcal{F}}_t)_{t \in [0,T]}, \bar{\mathbb{P}})\). Similarly, we will write \(\mathcal{V}^{p,q,r,\bar{\Omega}}_X\) for the space of stochastic controlled paths (in the sense of Definition 1) defined with respect to \((\bar{\Omega}, \bar{\mathcal{F}}, (\bar{\mathcal{F}}_t)_{t \in [0,T]}, \bar{\mathbb{P}})\).
Further, for any process \(Y = (Y_t)_{t \in [0,T]}\) defined on \((\hat{\Omega}, \hat{\mathcal{F}}, \hat{\mathbb{P}})\), and any two \(\mathcal{F}\)-measurable random times \(\tau_1, \tau_2\) on \((\Omega, \mathcal{F}, \mathbb{P})\) such that \(\tau_1 \leq \tau_2\) holds \(\mathbb{P}\)-almost surely, we define \(\|Y\|_{p,q,\infty, \llbracket \tau_1, \tau_2 \rrbracket, \bar{\Omega}}\) as the map from \(\Omega \to [0,\infty]\) given by \[\label{eq:32defn32p44q44infty32tau132tau232barOmega32norm} \Omega \ni \omega \mapsto \|Y(\omega, \cdot)\|_{p,q,\infty, [\tau_1(\omega), \tau_2(\omega)], \bar{\Omega}},\tag{16}\] and, with a slight abuse of notation, similarly define \(\sup_{s \in \llbracket \tau_1, \tau_2 \rrbracket} \|Y_s\|_{L^r(\bar{\Omega})}\) as the map given by \[\label{eq:32defn32sup32Ys32L94r32norm} \Omega \ni \omega \mapsto \sup_{s \in [\tau_1(\omega),\tau_2(\omega)]} \|Y_s(\omega, \cdot)\|_{L^r(\bar{\Omega})}.\tag{17}\]
Although it is not immediate that the maps defined in 16 and 17 are \(\mathcal{F}\)-measurable, and thus well-defined random variables, this is indeed the case, as stated precisely in the following lemma. The proofs of this and the following lemma are straightforward consequences of standard measurability and Fubini-type arguments, and are therefore omitted.
Lemma 3. Let \(p, q \in [1,\infty)\), \(r \in [q,\infty]\), \(\nu > 0\), and let \(\tau_1, \tau_2 \colon \Omega \to [0,T]\) be \(\mathcal{F}\)-measurable random times such that \(\tau_1 \leq \tau_2\). The following then hold.
If \(Z\) is an \(\hat{\mathcal{F}}\)-measurable random variable such that \(\|Z(\omega, \cdot)\|_{q,\infty,s,\bar{\Omega}} \leq \nu\) for \(\mathbb{P}\)-almost every \(\omega \in \Omega\), then \(\|Z\|_{q,r,s,\bar{\Omega}}\) is an \(\mathcal{F}\)-measurable random variable.
If \(Y = (Y_t)_{t \in [0,T]}\) is a \(\hat{\mathbb{P}}\)-almost surely càdlàg \(\hat{\mathcal{F}} \otimes \mathcal{B}([0,T])\)-measurable process, such that \(\|Y(\omega, \cdot)\|_{p,q,\infty,[0,T],\bar{\Omega}} \leq \nu\) for \(\mathbb{P}\)-almost every \(\omega \in \Omega\), then \(\|Y\|_{p,q,r,\llbracket \tau_1, \tau_2 \rrbracket,\bar{\Omega}}\) is an \(\mathcal{F}\)-measurable random variable.
Let \(Y\) be as in case (ii) above. Then \(\| 1 \wedge \sup_{t \in \llbracket \tau_1, \tau_2 \llbracket } |Y_t| \|_{L^q(\bar{\Omega})}\) is also \(\mathcal{F}\)-measurable.
Let \(Y\) be as in case (ii) above. If, in addition, \(\sup_{t \in [0,T]} \|Y_t(\omega, \cdot)\|_{L^{\infty}(\bar{\Omega})} < \infty\) for \(\mathbb{P}\)-almost every \(\omega \in \Omega\), then \(\sup_{t \in \llbracket \tau_1, \tau_2 \llbracket} \|Y_t\|_{L^{\infty}(\bar{\Omega})}\) is also \(\mathcal{F}\)-measurable.
Similarly, one can also show the following.
Lemma 4. For some \(p \in [2,3)\), \(q \in [2,\infty)\) and some constant \(\nu > 0\), suppose that \(X\) is an \(\mathcal{F}\otimes \mathcal{B}([0,T])\)-measurable process such that \(X(\omega) \in V^p\) for \(\mathbb{P}\)-almost every \(\omega \in \Omega\), and that \((Y,Y')\) is a pair of \(\hat{\mathcal{F}} \otimes \mathcal{B}([0,T])\)-measurable processes, such that \((Y(\omega, \cdot),Y'(\omega, \cdot)) \in \mathcal{V}^{p,q,\infty,\bar{\Omega}}_{X(\omega)}\) with \(\|(\bar{\mathbb{E}}_{\boldsymbol{\cdot}} R^Y)(\omega, \cdot)\|_{\frac{p}{2},\infty,[0,T],\bar{\Omega}} \leq \nu\) for \(\mathbb{P}\)-almost every \(\omega \in \Omega\).
Then, for any two \(\mathcal{F}\)-measurable random times \(\tau_1 \leq \tau_2\), and any \(r \in [q,\infty]\), we have that \(\|\bar{\mathbb{E}}_{\boldsymbol{\cdot}} R^Y\|_{\frac{p}{2},r,\llbracket \tau_1, \tau_2 \rrbracket,\bar{\Omega}}\) is an \(\mathcal{F}\)-measurable random variable.
For any two stopping times \(\tau_1 \leq \tau_2\), and any process \(X\), we let \(X^{(\tau_1, \tau_2-)} := (X - X^{\tau_1})^{\tau_2-}\), so that \(X^{(\tau_1,\tau_2-)} = 0\) on the event \(\{\tau_1 = \tau_2\}\), and on \(\{\tau_1 < \tau_2\}\) we have that \[\begin{align} X^{(\tau_1, \tau_2-)}_t := \begin{cases} 0 & \text{ if~~} t < \tau_1,\\ \delta X_{\tau_1, t} & \text{ if~~} \tau_1 \leq t < \tau_2,\\ \delta X_{\tau_1, \tau_2-} & \text{ if~~} t \geq \tau_2. \end{cases} \end{align}\] We note that this construction is analogous to that used in \(\alpha\)-slicing for classical Itô SDEs; cf. [44]. Further, if \(X\) is a semimartingale, we let \(\mathbf{X}^{(\tau_1, \tau_2-)} = (X^{(\tau_1, \tau_2-)}, \mathbb{X}^{(\tau_1, \tau_2-)})\) be the Itô rough path lift of \(X^{(\tau_1, \tau_2-)}\), i.e., \[\mathbb{X}^{(\tau_1, \tau_2-)}_{s,t} := \int_s^t \delta X^{(\tau_1, \tau_2-)}_{s,u-} \otimes \mathrm{d}X^{(\tau_1, \tau_2-)}_u\] for \((s,t) \in \Delta_{[0,T]}\).
Remark 5. We note that, if \(X\) is a semimartingale on \((\Omega, \mathcal{F}, \mathbb{P})\), and \(\mathbf{X}= (X,\mathbb{X})\) is its Itô rough path lift, and if \(\tau_1 \leq \tau_2\) are \(\mathcal{F}_t\)-stopping times, then for \(\mathbb{P}\)-almost any \(\omega \in \Omega\) and any \((s,t) \in \Delta_{[0,T]}\), we have that \[\mathbb{X}_{s,t}^{(\tau_1, \tau_2-)}(\omega) = \begin{cases} 0 , & \text{ if~~} 0 \leq t < \tau_1(\omega),\\ \mathbb{X}_{\tau_1(\omega) \vee s, t}(\omega), & \text{ if~~} \tau_1(\omega) \leq t < \tau_2(\omega),\\ \mathbb{X}_{\tau_1(\omega) \vee s, \tau_2(\omega)-}(\omega) & \text{ if~~} s < \tau_2(\omega) \leq t \leq T,\\ 0 & \text{ if~~} \tau_2(\omega) \leq s \leq T. \end{cases}\] Moreover, if \((Y,Y')\) is a pair of càdlàg \(\hat{\mathcal{F}}_t\)-adapted processes, then we see that the condition \[(Y(\omega, \cdot), Y'(\omega, \cdot)) \in \mathcal{V}_{X^{(\tau_1, \tau_2-)}(\omega)}^{p,q,\infty,\bar{\Omega}}([0,T])\] is equivalent to \[(Y(\omega, \cdot), Y'(\omega, \cdot)) \in \mathcal{V}^{p,q,\infty,\bar{\Omega}}_{X(\omega)}([\tau_1(\omega), \tau_2(\omega))).\] Thus, if these conditions hold for \(\mathbb{P}\)-almost every \(\omega \in \Omega\), then for almost every \(\omega \in \Omega\) and every \(t \in [\tau_1(\omega),\tau_2(\omega))\), the rough stochastic integrals \[\int_0^t Y_s(\omega, \cdot) \,\mathrm{d}\mathbf{X}^{(\tau_1, \tau_2-)}_s(\omega) = \int_{\tau_1(\omega)}^t Y_s(\omega, \cdot) \,\mathrm{d}\mathbf{X}_s(\omega)\] coincide. Similarly, for any (semi)martingale \(M\) and predictable process \(Z\), we have that \[\int_0^t Z_s \,\mathrm{d}M^{(\tau_1, \tau_2-)}_s = \bigg(\int_0^t Z_s \,\mathrm{d}M_s\bigg)^{(\tau_1, \tau_2-)}\] as well as \([M^{(\tau_1, \tau_2-)}]_t = [M]_t^{(\tau_1, \tau_2-)}\).
Let \(X\) be a process defined on \((\Omega, \mathcal{F}, \mathbb{P})\) such that \(X(\omega) \in V^p\) for \(\mathbb{P}\)-almost every \(\omega \in \Omega\). In the following we will consider pairs of processes \((Y,Y')\) defined on the product space \((\hat{\Omega},\hat{\mathcal{F}},\hat{\mathbb{P}})\), such that \((Y(\omega, \cdot), Y'(\omega, \cdot)) \in \mathcal{V}^{p,q,\infty,\bar{\Omega}}_{X^{(\tau_1, \tau_2-)}(\omega)}([\tau_1(\omega),\tau_2(\omega)])\) for \(\mathbb{P}\)-almost every \(\omega \in \Omega\), for some \(\mathcal{F}_t\)-stopping times \(\tau_1 \leq \tau_2\). For any two such pairs \((Y,Y')\), \((\widetilde{Y},\widetilde{Y}')\), and some \(\eta > 1\), we define \[\label{eq:32definition32of32metric32on32B95T32for32contraction} \begin{align} d_{\tau_1, \tau_2}^{\eta, \bar{\Omega}} \big( (Y,&Y'), (\widetilde{Y},\widetilde{Y}') \big) := \|Y' - \widetilde{Y}'\|_{p,q,\llbracket \tau_1, \tau_2 \rrbracket,\bar{\Omega}} + \Big\| 1 \wedge \sup_{u \in \llbracket \tau_1, \tau_2 \rrbracket} |Y'_u - \widetilde{Y}'_u| \Big\|_{L^q(\bar{\Omega})}\\ &+ \eta \Big( \|Y - \widetilde{Y}\|_{p,q,\llbracket \tau_1, \tau_2 \rrbracket,\bar{\Omega}} + \|\bar{\mathbb{E}}_{\boldsymbol{\cdot}} (R^Y - R^{\widetilde{Y}})\|_{\frac{p}{2},q,\llbracket \tau_1, \tau_2 \rrbracket,\bar{\Omega}} + \Big\| 1 \wedge \sup_{u \in \llbracket \tau_1, \tau_2 \rrbracket} |Y_u - \widetilde{Y}_u| \Big\|_{L^q(\bar{\Omega})} \Big). \end{align}\tag{18}\]
Remark 6. We note that this almost coincides with the metric defined in the proof of [22], except that here we additionally include the term \(\| 1 \wedge \sup_{u \in \llbracket \tau_1, \tau_2 \rrbracket} |Y'_u - \widetilde{Y}'_u| \|_{L^q(\bar{\Omega})}\). This term is included to ensure that limits under this metric are almost surely càdlàg, which is a necessary condition in Lemma 3 to ensure the measurability of \(\sup_{u \in \llbracket \tau_1, \tau_2 \llbracket} \|Y'_u\|_{L^\infty(\bar{\Omega})}\).
Let \(\tau_1 \leq \tau_2\) be \(\mathcal{F}_t\)-stopping times, and let \(\xi, \xi'\) be \(\hat{\mathcal{F}}_{\tau_1}\)-measurable random variables. Let \(X = (X_t)_{t \in [0,T]}\) be a process defined on \((\Omega, \mathcal{F}, \mathbb{P})\) which is adapted to \((\mathcal{F}_t)_{t \in [0,T]}\), and such that \(X(\omega) \in V^p\) for \(\mathbb{P}\)-almost every \(\omega \in \Omega\). For constants \(\nu_1, \nu_2 > 0\), we write \[\mathbf{B}_{\tau_1, \tau_2}(\nu_1, \nu_2, \xi, \xi')\] for the set of all pairs of processes \((Y,Y')\) such that
\(Y\) and \(Y'\) are both \(\hat{\mathcal{F}}_t\)-adapted and \(\hat{\mathbb{P}}\)-almost surely càdlàg,
\(\hat{\mathbb{P}}\)-almost surely, we have that \((Y,Y') = (0,0)\) on \(\llbracket 0, \tau_1 \llbracket\), with \((Y_{\tau_1},Y'_{\tau_1}) = (\xi,\xi')\), the processes \(Y\) and \(Y'\) are continuous at \(\tau_2\), and they are constant on \(\llbracket \tau_2, T \rrbracket\),
and we have that \((Y(\omega, \cdot), Y'(\omega, \cdot)) \in \mathcal{V}^{p,q,\infty,\bar{\Omega}}_{X^{(\tau_1, \tau_2-)}(\omega)}([\tau_1(\omega),\tau_2(\omega)])\), and satisfy \[\|Y'(\omega, \cdot)\|_{p,q,\infty,[0,T],\bar{\Omega}} \vee \sup_{u \in [0,T]} \|Y'_u(\omega, \cdot)\|_{L^\infty(\bar{\Omega})} \leq \nu_1\] and \[\|Y(\omega, \cdot)\|_{p,q,\infty,[0,T],\bar{\Omega}} \vee \|\bar{\mathbb{E}}_{\boldsymbol{\cdot}} R^Y(\omega, \cdot)\|_{\frac{p}{2},\infty,[0,T],\bar{\Omega}} \leq \nu_2\] for \(\mathbb{P}\)-almost every \(\omega \in \Omega\).
For any \(\eta > 1\) and any \((Y,Y'), (\widetilde{Y},\widetilde{Y}') \in \mathbf{B}_{\tau_1, \tau_2}(\nu_1, \nu_2, \xi, \xi')\), we define \[\mathbf{d}^\eta_{\tau_1, \tau_2} \big( (Y,Y'), (\widetilde{Y},\widetilde{Y}') \big) := \big\| d^{\eta, \bar{\Omega}}_{\tau_1, \tau_2} \big( (Y, Y'), (\widetilde{Y}, \widetilde{Y}') \big) \big\|_{L^\infty(\Omega)}.\]
Lemma 5. Let \(X = (X_t)_{t \in [0,T]}\) be a process defined on \((\Omega, \mathcal{F}, \mathbb{P})\) which is \(\mathcal{F}_t\)-adapted, and such that \(X(\omega) \in V^p\) for \(\mathbb{P}\)-almost every \(\omega \in \Omega\). Let \(\tau_1 \leq \tau_2\) be \(\mathcal{F}_t\)-stopping times, let \(\xi, \xi'\) be \(\hat{\mathcal{F}}_{\tau_1}\)-measurable random variables, and let \(\nu_1, \nu_2 > 0\) and \(\eta > 1\).
Then \(\mathbf{B}_{\tau_1, \tau_2}(\nu_1, \nu_2, \xi, \xi')\) and \(\mathbf{d}^\eta_{\tau_1, \tau_2}\) are both well-defined, and \((\mathbf{B}_{\tau_1, \tau_2}(\nu_1, \nu_2, \xi, \xi'), \mathbf{d}^\eta_{\tau_1, \tau_2})\) is a complete metric space.
To see that \(\mathbf{B}_{\tau_1, \tau_2}(\nu_1, \nu_2, \xi, \xi')\) and \(\mathbf{d}^\eta_{\tau_1, \tau_2}\) are well-defined is simply a matter of noting that all the norms involved are \(\mathcal{F}\)-measurable by Lemmas 3 and 4, so that it is then valid to take the \(L^\infty(\Omega)\) norm. Verifying that \((\mathbf{B}_{\tau_1, \tau_2}(\nu_1, \nu_2, \xi, \xi'), \mathbf{d}^\eta_{\tau_1, \tau_2})\) is a complete metric space then follows standard arguments, similar to the proofs of [22]. In particular, if a sequence is Cauchy with respect to \(\mathbf{d}^\eta_{\tau_1, \tau_2}\), then it is also Cauchy in the u.c.p. topology on \((\hat{\Omega}, \hat{\mathcal{F}}, \hat{\mathbb{P}})\), so that the existence and measurability of the limit follow immediately. The full proof of Lemma 5 is omitted for brevity.
The following result establishes consistency between Itô stochastic integrals and rough stochastic integrals against Itô rough path lifts of semimartingales. It will be unsurprising to readers familiar with rough analysis, but we nonetheless provide a proof in Appendix 6.3.
Proposition 7. Let \(p \in (2,3)\), \(q \in [2,\infty)\) and \(r \in [q,\infty]\), and let \((\Omega \times \bar{\Omega}, \mathcal{F}\otimes \bar{\mathcal{F}}, \mathbb{P}\otimes \bar{\mathbb{P}})\) be a product probability space which satisfies the usual conditions. Let \(X\) be a càdlàg semimartingale on \((\Omega, \mathcal{F}, \mathbb{P})\), and let \(\mathbf{X}= (X,\mathbb{X})\) be its Itô rough path lift, so that \(\mathbb{X}_{s,t} = \int_s^t \delta X_{s,u-} \otimes \mathrm{d}X_u\) for every \((s,t) \in \Delta_{[0,T]}\), and \(\mathbf{X}(\omega) \in \mathscr{V}^p\) for \(\mathbb{P}\)-almost every \(\omega \in \Omega\). We note that \(X\) is also a semimartingale on the product space (by identifying \(X_t(\omega, \bar{\omega}) = X_t(\omega)\) for all \((\omega,\bar{\omega}) \in \Omega \times \bar{\Omega}\) and \(t \in [0,T]\)). Let \(Y\) and \(Y'\) be adapted processes on \((\Omega \times \bar{\Omega}, \mathcal{F}\otimes \bar{\mathcal{F}}, \mathbb{P}\otimes \bar{\mathbb{P}})\) which are almost surely càdlàg, and suppose that \((Y(\omega, \cdot),Y'(\omega, \cdot)) \in \mathcal{V}_{X(\omega)}^{p,q,r,\bar{\Omega}}\) for \(\mathbb{P}\)-almost every \(\omega \in \Omega\).
Then, for \(\mathbb{P}\)-almost every \(\omega \in \Omega\), the processes \[\bigg( \int_0^\cdot Y_{u-} \,\mathrm{d}X_u \bigg)(\omega, \cdot) = \int_0^\cdot Y_u(\omega, \cdot) \,\mathrm{d}\mathbf{X}_u(\omega),\] are \(\bar{\mathbb{P}}\)-indistinguishable, where on the left-hand side is the Itô integral of \(Y_-\) against \(X\), and on the right-hand side is the rough stochastic integral of \((Y(\omega, \cdot),Y'(\omega, \cdot))\) against \(\mathbf{X}(\omega)\).
The following theorem is the main result of this section, and shows that the solution to a doubly stochastic DE is also the solution to the corresponding (random) rough SDE. The basic strategy is to construct a contraction mapping on the metric space in Lemma 5.
Remark 8. For simplicity of notation, here we present the result without including an integral against a random measure. However, using Assumption 1 and Lemma 16, it is straightforward to generalize the proof of Theorem 3 to show that the result also holds when an integral against a compensated random measure is included in the equation, as in 6 .
Theorem 3. Let \(p \in (2,3)\), \(q \in [2,\infty)\), \(b \in C^1_b\), \(\sigma \in C^1_b\) and \(f \in C^3_b\), and let \[(\hat{\Omega}, \hat{\mathcal{F}}, (\hat{\mathcal{F}}_t)_{t \in [0,T]}, \hat{\mathbb{P}}) = (\Omega \times \bar{\Omega}, \mathcal{F}\otimes \bar{\mathcal{F}}, (\mathcal{F}_t \otimes \bar{\mathcal{F}}_t)_{t \in [0,T]}, \mathbb{P}\otimes \bar{\mathbb{P}})\] be a product probability space which satisfies the usual conditions.
Let \(y_0\) be an \(\hat{\mathcal{F}}_0\)-measurable random variable such that \(\|y_0(\omega, \cdot)\|_{L^q(\bar{\Omega})} < \infty\) for \(\mathbb{P}\)-almost every \(\omega \in \Omega\). Let \(X\) be an \(\mathcal{F}_t\)-adapted càdlàg semimartingale on \((\Omega, \mathcal{F}, \mathbb{P})\), and let \(M \in V^p L^{q,\infty}(\bar{\Omega})\) be an \(\bar{\mathcal{F}}_t\)-adapted càdlàg martingale on \((\bar{\Omega}, \bar{\mathcal{F}}, \bar{\mathbb{P}})\).
Let \(Y\) be the unique strong solution to the SDE \[\label{eq:32the32doubly32stochastic32SDE} Y_t = y_0 + \int_0^t b(Y_s) \,\mathrm{d}s + \int_0^t \sigma(Y_{s-}) \,\mathrm{d}M_s + \int_0^t f(Y_{s-}) \,\mathrm{d}X_s, \qquad t \in [0,T],\tag{19}\] on \((\hat{\Omega}, \hat{\mathcal{F}}, \hat{\mathbb{P}})\), which in particular is \(\hat{\mathcal{F}}_t\)-adapted and has \(\hat{\mathbb{P}}\)-almost surely càdlàg sample paths.
Then, for \(\mathbb{P}\)-almost every \(\omega \in \Omega\), the pair \((Y(\omega, \cdot),f(Y(\omega, \cdot))) \in \mathcal{V}^{p,q,\infty,\bar{\Omega}}_{X(\omega)}\) is a stochastic controlled path relative to \(X(\omega)\), and \(Y(\omega, \cdot)\) is the solution to the rough SDE \[\label{eq:32the32corresponding32RSDE} Y_t(\omega, \cdot) = y_0(\omega, \cdot) + \int_0^t b(Y_s(\omega, \cdot)) \,\mathrm{d}s + \int_0^t \sigma(Y_{s-}(\omega, \cdot)) \,\mathrm{d}M_s + \int_0^t f(Y_s(\omega, \cdot)) \,\mathrm{d}\mathbf{X}_s(\omega)\tag{20}\] for \(t \in [0,T]\), where \(\mathbf{X}\) is the Itô rough path lift of \(X\).
Proof. We let \((\tau_i)_{i \in \mathbb{N}}\) be the sequence of \(\mathcal{F}_t\)-stopping times defined recursively by \(\tau_0 = 0\), and \[\tau_{i+1} = T \wedge \inf \big\{ t > \tau_i \, \big| \, (t - \tau_i) + \|M\|_{p,q,\infty,\llbracket \tau_i, t \rrbracket,\bar{\Omega}} + \|\mathbf{X}\|_{p,\llbracket \tau_i, t \rrbracket} \geq \alpha \big\}\] for each \(i \in \mathbb{N}\), where \(\alpha \in (0,1]\) is a constant which we will specify later.
Let us fix an \(i \in \mathbb{N}\). In particular, we then have that \[\label{eq:32small32interval32bound} (\tau_{i+1} - \tau_i) + \|M^{(\tau_i, \tau_{i+1}-)}\|_{p,q,\infty,\llbracket \tau_i, \tau_{i+1} \rrbracket,\bar{\Omega}} + \|\mathbf{X}^{(\tau_i, \tau_{i+1}-)}\|_{p,\llbracket \tau_i, \tau_{i+1} \rrbracket} \leq \alpha\tag{21}\] \(\mathbb{P}\)-almost surely. We also let \(\xi_i\) be an \(\hat{\mathcal{F}}_{\tau_i}\)-measurable random variable such that \(\xi_i(\omega, \cdot) \in L^q(\bar{\Omega})\) for \(\mathbb{P}\)-almost every \(\omega \in \Omega\).
For \((Y,Y') \in \mathbf{B}_{\tau_i,\tau_{i+1}}(\|f\|_{C^3_b}, 1, \xi_i, f(\xi_i))\), we let \[\begin{align} &\Psi(Y,Y') := \big(\psi(Y),\psi(Y)'\big)\\ &:= \bigg( \xi_i \mathbf{1}_{\llbracket \tau_i, T \rrbracket} + \int_{\tau_i \wedge \cdot}^{\tau_{i+1} \wedge \cdot} b(Y_u) \,\mathrm{d}u + \int_0^\cdot \sigma(Y_{u-}) \,\mathrm{d}M^{(\tau_i, \tau_{i+1}-)}_u + \int_0^\cdot f(Y_{u-}) \,\mathrm{d}X^{(\tau_i, \tau_{i+1}-)}_u, f(Y) \mathbf{1}_{\llbracket \tau_i, T \rrbracket} \bigg) \end{align}\] where the latter two integrals are both defined as Itô integrals on the product space.
We will show that the space \(\mathbf{B}_{\tau_i,\tau_{i+1}}(\|f\|_{C^3_b}, 1, \xi_i, f(\xi_i))\) is non-empty and invariant under the map \(\Psi\), and then that \(\Psi\) is a contraction on \(\mathbf{B}_{\tau_i,\tau_{i+1}}(\|f\|_{C^3_b}, 1, \xi_i, f(\xi_i))\) with respect to the metric \(\mathbf{d}^\eta_{\tau_i,\tau_{i+1}}\), for suitable choices of \(\alpha \in (0,1]\) and \(\eta > 1\).
We first note that, if we set \[(Y_t,Y'_t) = \big( \big(\xi_i + f(\xi_i) X^{(\tau_i, \tau_{i+1}-)}_t\big) \mathbf{1}_{\llbracket \tau_i, T \rrbracket}, f(\xi_i) \mathbf{1}_{\llbracket \tau_i, T \rrbracket} \big)\] for all \(t \in [0,T]\), then, provided that \(\alpha \leq \|f\|_{C^3_b}^{-1}\), we have that \((Y,Y') \in \mathbf{B}_{\tau_i,\tau_{i+1}}(\|f\|_{C^3_b}, 1, \xi_i, f(\xi_i))\), so that the space \(\mathbf{B}_{\tau_i,\tau_{i+1}}(\|f\|_{C^3_b}, 1, \xi_i, f(\xi_i))\) is non-empty.
In the following we will recall some estimates from the proof of [22]. These are written in terms of a function \(\Phi\), which, given a time interval \([0,T]\), an initial value \(\xi\), a rough path \(\mathbf{Z}= (Z,\mathbb{Z})\) and stochastic controlled path \((Y,Y') \in \mathcal{V}^{p,q,\infty}_Z\), is defined by \[\Phi_{\mathbf{Z}}(Y,Y') := \big( \phi_{\mathbf{Z}}(Y), \phi_{\mathbf{Z}}(Y)' \big) := \bigg( \xi + \int_0^\cdot b(Y_u) \,\mathrm{d}u + \int_0^\cdot \sigma(Y_{u-}) \,\mathrm{d}M_u + \int_0^\cdot f(Y_u) \,\mathrm{d}\mathbf{Z}_u, f(Y) \bigg).\]
By Proposition 7, we have that, for \(\mathbb{P}\)-almost every \(\omega \in \Omega\), \[\bigg( \int_0^\cdot f(Y_{u-}) \,\mathrm{d}X^{(\tau_i, \tau_{i+1}-)}_u \bigg) (\omega, \cdot) = \int_0^\cdot f(Y_u(\omega, \cdot)) \,\mathrm{d}\mathbf{X}^{(\tau_i, \tau_{i+1}-)}_u(\omega)\] \(\bar{\mathbb{P}}\)-almost surely, where on the right-hand side is the rough stochastic integral of the stochastic controlled path \((f(Y(\omega, \cdot)), \mathrm{D}f(Y(\omega, \cdot)) Y'(\omega, \cdot))\) with respect to the rough path \(\mathbf{X}^{(\tau_i, \tau_{i+1}-)}(\omega)\).
By considering convergence of the Riemann sums \(\lim_{|\mathcal{P}| \to 0} \sum_{[u,v] \in \mathcal{P}} \sigma(Y_u) \delta M^{(\tau_i, \tau_{i+1}-)}_{u,v}\), it is also straightforward to see that, for \(\mathbb{P}\)-almost every \(\omega \in \Omega\), \[\bigg( \int_0^\cdot \sigma(Y_{u-}) \,\mathrm{d}M^{(\tau_i, \tau_{i+1}-)}_u \bigg) (\omega, \cdot) = \int_0^\cdot \sigma(Y_{u-}(\omega, \cdot)) \,\mathrm{d}M^{(\tau_i(\omega), \tau_{i+1}(\omega)-)}_u\] \(\bar{\mathbb{P}}\)-almost surely, noting that \(M^{(\tau_i(\omega), \tau_{i+1}(\omega)-)}\) is a martingale with respect to \((\bar{\mathcal{F}}_t)_{t \in [0,T]}\). This means that, for \(\mathbb{P}\)-almost every \(\omega \in \Omega\), \[\begin{align} \psi(Y)(\omega, \cdot) &= \xi_i \mathbf{1}_{\llbracket \tau_i, T \rrbracket} + \int_{\tau_i(\omega) \wedge \cdot}^{\tau_{i+1}(\omega) \wedge \cdot} b(Y_u(\omega, \cdot)) \,\mathrm{d}u + \int_0^\cdot \sigma(Y_{u-}(\omega, \cdot)) \,\mathrm{d}M^{(\tau_i(\omega), \tau_{i+1}(\omega)-)}_u\\ &\quad \, + \int_0^\cdot f(Y_u(\omega, \cdot)) \,\mathrm{d}\mathbf{X}^{(\tau_i, \tau_{i+1}-)}_u(\omega)\\ &= \phi_{\mathbf{X}(\omega)}(Y(\omega, \cdot)) \end{align}\] \(\bar{\mathbb{P}}\)-almost surely, and it is also clear that \(\psi(Y)'(\omega, \cdot) = f(Y(\omega, \cdot)) = \phi_{\mathbf{X}(\omega)}(Y(\omega, \cdot))'\).
Invariance: To see that \(\mathbf{B}_{\tau_i,\tau_{i+1}}(\|f\|_{C^3_b}, 1, \xi_i, f(\xi_i))\) is invariant under \(\Psi\), we can now simply apply the estimates derived in an analogous context in Step 1 of the proof of [22]. Specifically, for \(\mathbb{P}\)-almost every \(\omega \in \Omega\), we have that \[\begin{align} &\|\psi(Y)'(\omega, \cdot)\|_{p,q,\infty,[\tau_i(\omega),\tau_{i+1}(\omega)],\bar{\Omega}} \vee \sup_{s \in [\tau_i(\omega),\tau_{i+1}(\omega)]}\|\psi(Y)'_s(\omega, \cdot)\|_{L^\infty(\bar{\Omega})}\\ &= \|\phi_{\mathbf{X}(\omega)}(Y(\omega, \cdot))'\|_{p,q,\infty,[\tau_i(\omega),\tau_{i+1}(\omega)],\bar{\Omega}} \vee \sup_{s \in [\tau_i(\omega),\tau_{i+1}(\omega)]}\|\phi_{\mathbf{X}(\omega)}(Y(\omega, \cdot))'_s\|_{L^\infty(\bar{\Omega})} \leq \|f\|_{C^3_b}, \end{align}\] and \[\begin{align} &\|\psi(Y)(\omega, \cdot)\|_{p,q,\infty,[\tau_i(\omega),\tau_{i+1}(\omega)],\bar{\Omega}} \vee \|\bar{\mathbb{E}}_{\boldsymbol{\cdot}} R^{\psi(Y)(\omega, \cdot)}\|_{\frac{p}{2},\infty,[\tau_i(\omega),\tau_{i+1}(\omega)],\bar{\Omega}}\\ &= \|\phi_{\mathbf{X}(\omega)}(Y(\omega, \cdot))\|_{p,q,\infty,[\tau_i(\omega),\tau_{i+1}(\omega)],\bar{\Omega}} \vee \|\bar{\mathbb{E}}_{\boldsymbol{\cdot}} R^{\phi_{\mathbf{X}(\omega)}(Y(\omega, \cdot))}\|_{\frac{p}{2},\infty,[\tau_i(\omega),\tau_{i+1}(\omega)],\bar{\Omega}}\\ &\leq C_1 \Big((\tau_{i+1}(\omega) - \tau_i(\omega)) + \|M^{(\tau_i(\omega),\tau_{i+1}(\omega)-)}\|_{p,q,\infty,[\tau_i(\omega),\tau_{i+1}(\omega)],\bar{\Omega}}\\ &\qquad \quad + \|\mathbf{X}^{(\tau_i(\omega),\tau_{i+1}(\omega)-)}(\omega)\|_{p,[\tau_i(\omega),\tau_{i+1}(\omega)]}\Big), \end{align}\] where the constant \(C_1\) depends only on \(p, q, \|b\|_{C^1_b}, \|\sigma\|_{C^1_b}\) and \(\|f\|_{C^3_b}\).9 Provided that we choose \(\alpha \leq \frac{1}{C_1}\), it then follows from 21 that \[\|\psi(Y)(\omega, \cdot)\|_{p,q,\infty,[\tau_i(\omega),\tau_{i+1}(\omega)],\bar{\Omega}} \vee \|\bar{\mathbb{E}}_{\boldsymbol{\cdot}} R^{\psi(Y)(\omega, \cdot)}\|_{\frac{p}{2},\infty,[\tau_i(\omega),\tau_{i+1}(\omega)],\bar{\Omega}} \leq 1.\] Since these estimates hold for \(\mathbb{P}\)-almost every \(\omega \in \Omega\), it is then clear that \[\big\| \|\psi(Y)'\|_{p,q,\infty,\llbracket \tau_i, \tau_{i+1} \rrbracket,\bar{\Omega}} \big\|_{L^\infty(\Omega)} \vee \Big\| \sup_{s \in \llbracket \tau_i, \tau_{i+1} \rrbracket} \|\psi(Y)'_s\|_{L^\infty(\bar{\Omega})} \Big\|_{L^\infty(\Omega)} \leq \|f\|_{C^3_b}\] and \[\big\| \|\psi(Y)\|_{p,q,\infty,\llbracket \tau_i, \tau_{i+1} \rrbracket,\bar{\Omega}} \big\|_{L^\infty(\Omega)} \vee \big\| \|\bar{\mathbb{E}}_{\boldsymbol{\cdot}} R^{\psi(Y)}\|_{\frac{p}{2},\infty,\llbracket \tau_i, \tau_{i+1} \rrbracket,\bar{\Omega}} \big\|_{L^\infty(\Omega)} \leq 1.\] It follows that \(\Psi(Y,Y') \in \mathbf{B}_{\tau_i,\tau_{i+1}}(\|f\|_{C^3_b}, 1, \xi_i, f(\xi_i))\), and hence that \(\mathbf{B}_{\tau_i,\tau_{i+1}}(\|f\|_{C^3_b}, 1, \xi_i, f(\xi_i))\) is invariant under the map \(\Psi\).
Contraction: Similarly, to see that \(\Psi\) is a contraction on \(\mathbf{B}_{\tau_i,\tau_{i+1}}(\|f\|_{C^3_b}, 1, \xi_i, f(\xi_i))\) with respect to \(\mathbf{d}^\eta_{\tau_i,\tau_{i+1}}\), we apply the estimates derived in an analogous setting in Step 2 of the proof of [22] (which rely on the invariance established above). Specifically, for \((Y,Y'), (\widetilde{Y}, \widetilde{Y}') \in \mathbf{B}_{\tau_i,\tau_{i+1}}(\|f\|_{C^3_b}, 1, \xi_i, f(\xi_i))\), and \(\mathbb{P}\)-almost any \(\omega \in \Omega\), we have that \[\begin{align} &d^{\eta,\bar{\Omega}}_{\tau_i,\tau_{i+1}} \big(\Psi(Y,Y')(\omega, \cdot), \Psi(\widetilde{Y},\widetilde{Y}')(\omega, \cdot)\big)\\ &= d^{\eta,\bar{\Omega}}_{\tau_i,\tau_{i+1}} \big( \Phi_{\mathbf{X}(\omega)}\big(Y(\omega, \cdot), Y'(\omega, \cdot)\big), \Phi_{\mathbf{X}(\omega)}\big(\widetilde{Y}(\omega, \cdot), \widetilde{Y}'(\omega, \cdot)\big) \big)\\ &\leq C_2 \Big( \Big( \|Y(\omega, \cdot) - \widetilde{Y}(\omega, \cdot)\|_{p,q,[\tau_i(\omega),\tau_{i+1}(\omega)],\bar{\Omega}} + \Big\| 1 \wedge \sup_{u \in [\tau_i(\omega),\tau_{i+1}(\omega)]} |Y_u(\omega, \cdot) - \widetilde{Y}_u(\omega, \cdot)| \Big\|_{L^q(\bar{\Omega})} \Big)\\ &\quad + \eta \Big( \|Y(\omega, \cdot) - \widetilde{Y}(\omega, \cdot)\|_{p,q,[\tau_i(\omega),\tau_{i+1}(\omega)],\bar{\Omega}} + \|Y'(\omega, \cdot) - \widetilde{Y}'(\omega, \cdot)\|_{p,q,[\tau_i(\omega),\tau_{i+1}(\omega)],\bar{\Omega}}\\ &\qquad + \|\bar{\mathbb{E}}_{\boldsymbol{\cdot}} (R^{Y(\omega, \cdot)} - R^{\widetilde{Y}(\omega, \cdot)})\|_{\frac{p}{2},q,[\tau_i(\omega),\tau_{i+1}(\omega)],\bar{\Omega}} + \Big\| 1 \wedge \sup_{u \in [\tau_i(\omega),\tau_{i+1}(\omega)]} |Y_u(\omega, \cdot) - \widetilde{Y}_u(\omega, \cdot)| \Big\|_{L^q(\bar{\Omega})} \Big)\\ &\qquad \;\times \Big( (\tau_{i+1}(\omega) - \tau_i(\omega)) + \|M^{(\tau_i(\omega),\tau_{i+1}(\omega)-)}\|_{p,q,\infty,[\tau_i(\omega),\tau_{i+1}(\omega)],\bar{\Omega}}\\ &\qquad \qquad + \|\mathbf{X}^{(\tau_i(\omega),\tau_{i+1}(\omega)-)}(\omega)\|_{p,[\tau_i(\omega),\tau_{i+1}(\omega)]} \Big) \Big), \end{align}\] where the constant \(C_2 > \frac{1}{2}\) depends only on \(p, q, \|b\|_{C^1_b}, \|\sigma\|_{C^1_b}\) and \(\|f\|_{C^3_b}\).10
By choosing \(\eta = 2 C_2 > 1\) and \(\alpha \leq \frac{1}{4 C_2^2}\), it then follows from 21 that \[d^{\eta,\bar{\Omega}}_{\tau_i,\tau_{i+1}} \big(\Psi(Y,Y')(\omega, \cdot), \Psi(\widetilde{Y},\widetilde{Y}')(\omega, \cdot)\big) \leq \frac{\eta + 1}{2\eta} d^{\eta,\bar{\Omega}}_{\tau_i,\tau_{i+1}} \big( (Y,Y')(\omega, \cdot), (\widetilde{Y},\widetilde{Y}')(\omega, \cdot)\big).\] Since this inequality holds for \(\mathbb{P}\)-almost every \(\omega \in \Omega\), we then have that \[\mathbf{d}^\eta_{\tau_i,\tau_{i+1}} \big(\Psi(Y,Y'), \Psi(\widetilde{Y},\widetilde{Y}')\big) \leq \frac{\eta + 1}{2\eta} \mathbf{d}^\eta_{\tau_i,\tau_{i+1}} \big( (Y,Y'), (\widetilde{Y},\widetilde{Y}')\big),\] so that \(\Psi\) is indeed a contraction on \(\mathbf{B}_{\tau_i,\tau_{i+1}}(\|f\|_{C^3_b}, 1, \xi_i, f(\xi_i))\) with respect to \(\mathbf{d}^\eta_{\tau_i,\tau_{i+1}}\). Since, by Lemma 5, this is a complete metric space, it follows from the Banach fixed-point theorem that there exists a unique fixed point \((Y,Y') \in \mathbf{B}_{\tau_i,\tau_{i+1}}(\|f\|_{C^3_b}, 1, \xi_i, f(\xi_i))\) of \(\Psi\).
In particular, \(\hat{\mathbb{P}}\)-almost surely, whenever \(t \in \llbracket \tau_i, \tau_{i+1} \llbracket\), we have that \[Y_t = \xi_i + \int_{\tau_i}^t b(Y_u) \,\mathrm{d}u + \int_{\tau_i}^t \sigma(Y_{u-}) \,\mathrm{d}M_u + \int_{\tau_i}^t f(Y_{u-}) \,\mathrm{d}X_u,\] and, recalling Remark 5, for \(\mathbb{P}\)-almost every \(\omega \in \Omega\) and any \(t \in [\tau_i(\omega), \tau_{i+1}(\omega))\), \[\begin{align} Y_t(\omega, \cdot) = \xi_i(\omega, \cdot) + \int_{\tau_i}^t b(Y_u(\omega, \cdot)) \,\mathrm{d}u + \int_{\tau_i}^t \sigma(Y_{u-}(\omega, \cdot)) \,\mathrm{d}M_u + \int_{\tau_i}^t f(Y_u(\omega, \cdot)) \,\mathrm{d}\mathbf{X}_u(\omega) \end{align}\] holds \(\bar{\mathbb{P}}\)-almost surely, so that \(Y\) is the solution to both the doubly SDE 19 and the rough SDE 20 on the interval \(\llbracket \tau_i, \tau_{i+1} \llbracket\), with the initial condition \(Y_{\tau_i} = \xi_i\).
Global solution: Now let \((\xi_i)_{i \in \mathbb{N}}\) be a sequence of random variables such that, for each \(i \in \mathbb{N}\), \(\xi_i\) is \(\hat{\mathcal{F}}_{\tau_i}\)-measurable and \(\xi(\omega, \cdot) \in L^q(\bar{\Omega})\) for \(\mathbb{P}\)-almost every \(\omega \in \Omega\). Of course, the argument above is valid on each of the intervals \(\llbracket \tau_i, \tau_{i+1} \llbracket\) for \(i \in \mathbb{N}\). Thus, for each \(i \in \mathbb{N}\), we obtain a solution \(Y\) on the interval \(\llbracket \tau_i, \tau_{i+1} \llbracket\), with the initial condition \(Y_{\tau_i} = \xi_i\).
We then extend this to a solution on \(\llbracket \tau_i, \tau_{i+1} \rrbracket\) by introducing the jump at time \(\tau_{i+1}\). That is, given the solution \(Y\) on \(\llbracket \tau_i, \tau_{i+1} \llbracket\), we let \[Y_{\tau_{i+1}} = Y_{\tau_{i+1}-} + \sigma(Y_{\tau_{i+1}-}) \Delta M_{\tau_{i+1}} + f(Y_{\tau_{i+1}-}) \Delta X_{\tau_{i+1}},\] which is consistent with the canonical jump structure of both SDEs and rough SDEs, noting in particular that \(\Delta \mathbb{X}_{\tau_{i+1}} = 0\) for any Itô rough path lift.
We then obtain a global solution \(Y\) by simply matching each initial value \(\xi_i\) with the terminal value from the previous subinterval. That is, we let \(\xi_0 = y_0\), and for each \(i \geq 1\), we let \(\xi_i = Y_{\tau_i}\), where \(Y\) is the solution obtained above on the interval \(\llbracket \tau_{i-1}, \tau_i \rrbracket\). Since, for \(\mathbb{P}\)-almost every \(\omega \in \Omega\), we have that \(\tau_i(\omega) = T\) for sufficiently large \(i\), we see that this defines a process \(Y\) on \([0,T]\), which is the unique solution to both the SDE in 19 and the rough SDE in 20 . ◻
In light of Theorem 3, one may ask whether the stability of rough SDE solutions with respect to their driving noise carry over to the corresponding doubly stochastic setting. Indeed, the following result establishes a novel continuity estimate for solutions to doubly stochastic DEs.
Proposition 9. Let \(p \in (2,3)\), \(q \in [2,\infty)\), and let \(m, n, r \in [1,\infty]\) such that \(\frac{1}{n} + \frac{1}{r} = \frac{1}{m}\). Let \(b \in C^1_b\), \(\sigma \in C^1_b\) and \(f \in C^3_b\), and let \[(\hat{\Omega}, \hat{\mathcal{F}}, (\hat{\mathcal{F}}_t)_{t \in [0,T]}, \hat{\mathbb{P}}) = (\Omega \times \bar{\Omega}, \mathcal{F}\otimes \bar{\mathcal{F}}, (\mathcal{F}_t \otimes \bar{\mathcal{F}}_t)_{t \in [0,T]}, \mathbb{P}\otimes \bar{\mathbb{P}})\] be a product probability space which satisfies the usual conditions.
Let \(y_0, \widetilde{y}_0\) be \(\hat{\mathcal{F}}_0\)-measurable random variables such that \(\|y_0\|_{L^q(\bar{\Omega})}, \|\tilde{y}_0\|_{L^q(\bar{\Omega})} \in L^r(\Omega)\). Let \(X\) and \(\widetilde{X}\) be \(\mathcal{F}_t\)-adapted càdlàg semimartingales on \((\Omega, \mathcal{F}, \mathbb{P})\), such that their Itô rough path lifts \(\mathbf{X}, \widetilde{\mathbf{X}}\) satisfy \(\|\mathbf{X}\|_{p,[0,T]}, \|\widetilde{\mathbf{X}}\|_{p,[0,T]} \in L^{2np}(\Omega) \cap L^r(\Omega)\), and let \(M, \widetilde{M}\in V^p L^{q,\infty}(\bar{\Omega})\) be \(\bar{\mathcal{F}}_t\)-adapted càdlàg martingales on \((\bar{\Omega}, \bar{\mathcal{F}}, \bar{\mathbb{P}})\), and suppose that the norms \(\|\|\mathbf{X}\|_{p,[0,T]}\|_{L^{2np}(\Omega)}\), \(\|\|\widetilde{\mathbf{X}}\|_{p,[0,T]}\|_{L^{2np}(\Omega)}\), \(\|M\|_{p,q,\infty,[0,T],\bar{\Omega}}\) and \(\|\widetilde{M}\|_{p,q,\infty,[0,T],\bar{\Omega}}\) are all bounded by a constant \(L > 0\).
Let \(Y\) and \(\widetilde{Y}\) be the unique strong solutions to the SDE 19 with data \((y_0,M,X)\) and \((\widetilde{y}_0,\widetilde{M},\widetilde{X})\) respectively. We then have that \[\begin{align} \big\|&\|Y - \widetilde{Y}\|_{p,q,[0,T], \bar{\Omega}}\big\|_{L^m(\Omega)} + \big\|\|Y' - \widetilde{Y}'\|_{p,q,[0,T],\bar{\Omega}}\big\|_{L^m(\Omega)} + \big\|\|\mathbb{E}_{\boldsymbol{\cdot}} (R^Y - R^{\widetilde{Y}})\|_{\frac{p}{2},q,[0,T],\bar{\Omega}}\big\|_{L^m(\Omega)}\\ &\leq C \Big(\big\|\|y_0 - \widetilde{y}_0\|_{L^q(\bar{\Omega})}\big\|_{L^{r}(\Omega)} + \|M - \widetilde{M}\|_{p,q,[0,T],\bar{\Omega}} + \big\|\|\mathbf{X}- \widetilde{\mathbf{X}}\|_{p,[0,T]}\big\|_{L^{r}(\Omega)}\Big), \end{align}\] where the constant \(C\) depends only on \(p, q, \|b\|_{C^1_b}, \|\sigma\|_{C^1_b}, \|f\|_{C^3_b}, T\) and \(L\).
Proof. We let \(w\) be the control given by \[w(s,t) := (t - s) + \|M\|_{p,q,\infty,[s,t],\bar{\Omega}}^p + \|\widetilde{M}\|_{p,q,\infty,[s,t],\bar{\Omega}}^p + \|\mathbf{X}\|_{p,[s,t]}^p + \|\widetilde{\mathbf{X}}\|_{p,[s,t]}^p\] for \((s,t) \in \Delta_{[0,T]}\). In the following, we will consider intervals \([s,t]\) such that \(w(s,t) \leq 1\), meaning in particular that \(\|\mathbf{X}\|_{p,[s,t]}, \|\widetilde{\mathbf{X}}\|_{p,[s,t]} \leq 1\).
By Theorem 3, we know that \(Y\) and \(\widetilde{Y}\) are the solutions to the rough SDE in 20 with data \((y_0,M,\mathbf{X})\) and \((\widetilde{y}_0,\widetilde{M},\widetilde{\mathbf{X}})\) respectively. It was established in Step 3 of the proof of [22] that there exists a constant \(\varepsilon\in (0,1]\), which depends only on \(p, q, \|b\|_{C^1_b}, \|\sigma\|_{C^1_b}\) and \(\|f\|_{C^3_b}\), such that, for every \((s,t) \in \Delta_{[0,T]}\) with \(w(s,t) \leq \varepsilon\), we have the local estimate \[\label{eq:32local32robustness32result} \begin{align} \|&Y - \widetilde{Y}\|_{p,q,[s,t],\bar{\Omega}} + \|Y' - \widetilde{Y}'\|_{p,q,[s,t],\bar{\Omega}} + \|\mathbb{E}_{\boldsymbol{\cdot}} (R^Y - R^{\widetilde{Y}})\|_{\frac{p}{2},q,[s,t],\bar{\Omega}}\\ &\lesssim \big( \|Y_s - \widetilde{Y}_s\|_{L^q(\bar{\Omega})} + \|M - \widetilde{M}\|_{p,q,[s,t],\bar{\Omega}} + \|\mathbf{X}- \widetilde{\mathbf{X}}\|_{p,[s,t]} \big), \end{align}\tag{22}\] where the implicit multiplicative constant also depends only on \(p, q, \|b\|_{C^1_b}, \|\sigma\|_{C^1_b}\) and \(\|f\|_{C^3_b}\). In particular, by taking \(\varepsilon\leq 1\), we ensure that \(\|\mathbf{X}\|_{p,[s,t]}, \|\widetilde{\mathbf{X}}\|_{p,[s,t]} \leq 1\), so that these constants may be chosen to not depend on \(\mathbf{X}\) or \(\widetilde{\mathbf{X}}\).
To then extend this to a global estimate, we take a partition \(\{t_i\}_{i=0}^K\) of the interval \([0,T]\) such that \(w(t_i,t_{i+1}-) \leq \varepsilon\), so that 22 holds on \([t_i,t_{i+1})\) for each \(i\). By the superadditivity of the control \(w\), we may choose this partition such that \[\label{eq:32bound32on32n32without32X32and32tX} K \lesssim 1 + \|\mathbf{X}\|_{p,[0,T]}^p + \|\widetilde{\mathbf{X}}\|_{p,[0,T]}^p,\tag{23}\] where the multiplicative constant depends only on \(p, q, \|b\|_{C^1_b}, \|\sigma\|_{C^1_b}, \|f\|_{C^3_b}\) and on the norms \(\|M\|_{p,q,\infty,[0,T],\bar{\Omega}}\) and \(\|\widetilde{M}\|_{p,q,\infty,[0,T],\bar{\Omega}}\). Using the canonical jump structure of \(Y\) and \(\widetilde{Y}\), we have that \[\|Y - \widetilde{Y}\|_{p,q,[t_{i-1},t_i],\bar{\Omega}} \lesssim \|Y_{t_{i-1}} - \widetilde{Y}_{t_{i-1}}\|_{L^q(\bar{\Omega})} + \|M - \widetilde{M}\|_{p,q,[t_{i-1},t_i],\bar{\Omega}} + \|\mathbf{X}- \widetilde{\mathbf{X}}\|_{p,[t_{i-1},t_i]}.\] Since \(\|Y_{t_{i-1}} - \widetilde{Y}_{t_{i-1}}\|_{L^q(\bar{\Omega})} \leq \|Y_{t_{i-2}} - \widetilde{Y}_{t_{i-2}}\|_{L^q(\bar{\Omega})} + \|Y - \widetilde{Y}\|_{p,q,[t_{i-2},t_{i-1}],\bar{\Omega}}\) for each \(i\), we then deduce that \[\|Y - \widetilde{Y}\|_{p,q,[t_{i-1},t_i],\bar{\Omega}} \lesssim i \big(\|y_0 - \widetilde{y}_0\|_{L^q(\bar{\Omega})} + \|M - \widetilde{M}\|_{p,q,[0,t_i],\bar{\Omega}} + \|\mathbf{X}- \widetilde{\mathbf{X}}\|_{p,[0,t_i]}\big).\] We then have \[\begin{align} \|Y - \widetilde{Y}\|_{p,q,[0,T],\bar{\Omega}} &\leq K^{\frac{p-1}{p}} \bigg( \sum_{i=0}^{K-1} \|Y - \widetilde{Y}\|_{p,q,[t_i,t_{i+1}],\bar{\Omega}}^p \bigg)^{\frac{1}{p}}\\ &\lesssim K^2 \big( \|y_0 - \widetilde{y}_0\|_{L^q(\bar{\Omega})} + \|M - \widetilde{M}\|_{p,q,[0,T],\bar{\Omega}} + \|\mathbf{X}- \widetilde{\mathbf{X}}\|_{p,[0,T]} \big), \end{align}\] where again the multiplicative constant is independent of \(\mathbf{X}\) and \(\widetilde{\mathbf{X}}\). Thus, applying Hölder’s inequality in the case when \(n, r < \infty\), we have that \[\begin{align} &\big\|\|Y - \widetilde{Y}\|_{p,q,[0,T],\bar{\Omega}}\big\|_{L^m(\Omega)}\\ &\lesssim \|K^2\|_{L^n(\Omega)} \Big( \big\|\|y_0 - \widetilde{y}_0\|_{L^q(\bar{\Omega})}\big\|_{L^r(\Omega)} + \|M - \widetilde{M}\|_{p,q,[0,T],\bar{\Omega}} + \big\|\|\mathbf{X}- \widetilde{\mathbf{X}}\|_{p,[0,T]}\big\|_{L^r(\Omega)} \Big), \end{align}\] where, by 23 , we see that \(\|K^2\|_{L^n(\Omega)} \lesssim 1 + \|\|\mathbf{X}\|_{p,[0,T]}\|_{L^{2np}(\Omega)}^{2p} + \|\|\widetilde{\mathbf{X}}\|_{p,[0,T]}\|_{L^{2np}(\Omega)}^{2p}\). An identical argument gives the same bound for \(\|Y' - \widetilde{Y}'\|_{p,q,[0,T],\bar{\Omega}}\) and for \(\|\mathbb{E}_{\boldsymbol{\cdot}} (R^Y - R^{\widetilde{Y}})\|_{\frac{p}{2},q,[0,T],\bar{\Omega}}\). ◻
Integrability properties of random rough paths and associated RDEs have drawn significant interest; see, e.g., [53], [54], [55], [56] and [57]. In particular, the existence of exponential moments for solutions to rough SDEs has been crucial for applications to robust stochastic filtering, as seen in [13] and [17]. A particularly useful object in the study of exponential moments of solutions to RDEs turned out to be an object, typically denoted by \(N_{\alpha, [s,t]}(w)\) for a continuous control \(w\), first introduced in [53], which counts how many times the control exceeds a given level \(\alpha > 0\) on an interval \([s,t]\). This has been shown to often exhibit better tail behaviour than \(\|\mathbf{X}\|_{p,[0,T]}^p\) for the rough path lift of a semimartingale; see, e.g., [55].
In [45], the author discusses a relevant class of processes with vanishing mean oscillation, denoted by \(\mathrm{VMO}^{p\mathrm{-var}}\), and their connection to rough SDEs. In particular, a John–Nirenberg-type inequality provides an explicit bound on the exponential moments of such processes (see [45]), and the existence of exponential moments is established for the solutions to rough SDEs with continuous driving noise. In this section, we extend these results to discontinuous processes, and establish exponential moments of rough stochastic integrals and of solutions to rough SDEs with jumps, as required for our subsequent application to robust filtering. To this end, we first extend the notion of \(\mathrm{VMO}^{p\mathrm{-var}}\) processes to more general \(\mathrm{BMO}^{p\mathrm{-var}}\) ones, and relate the corresponding bounds to a generalization of \(N_{\alpha,[s,t]}(w)\) to discontinuous controls. This both establishes the desired exponential moments for rough stochastic integrals, and will also allow us to establish corresponding moments in a doubly stochastic setting in Section 5.
For convenience, we recall the notion of a BMO process, along with some relevant notation as introduced in [45].
Definition 10. Given a càdlàg adapted process \(V = (V_t)_{t \in [0,T]}\), we define its modulus of mean variation over an interval \([s,t] \subseteq [0,T]\) by \[\rho_{s,t}(V) := \sup_{s \leq \sigma \leq \tau \leq t} \big\|\mathbb{E}_{\sigma} [|\delta V_{\sigma-,\tau}|]\big\|_{L^\infty},\] where the supremum is taken over all stopping times \(\sigma, \tau\) such that \(s \leq \sigma \leq \tau \leq t\). The process \(V\) is said to be of bounded mean oscillation (BMO) if \(\rho_{0,T}(V) < \infty\).
Definition 11. For \(p \in [1,\infty)\), we write \(\mathrm{BMO}^{p\mathrm{-var}}\) for the class of càdlàg adapted processes \(V = (V_t)_{t \in [0,T]}\) such that \[\|V\|_{\mathrm{BMO}^{p\text{-var}},[0,T]} := \bigg(\sup_{\mathcal{P}\subset [0,T]} \sum_{[s,t] \in \mathcal{P}} \rho_{s,t}(V)^p\bigg)^{\frac{1}{p}} < \infty.\]
The following result is a generalization of [45] to the càdlàg setting.
Proposition 12. Let \(p, q \in [1,\infty)\). Let \(V\) be a càdlàg adapted process such that \(V_0 \in L^q\), and define \(\Gamma = (\Gamma_{s,t})_{(s,t) \in \Delta_{[0,T]}}\) by \[\label{eq:32defn32Gamma95st} \Gamma_{s,t} := \sup_{u \in [s,t]} |\Delta V_u|\qquad{(1)}\] for each \((s,t) \in \Delta_{[0,T]}\). Then there exists a constant \(C > 0\), which depends only on \(q\), such that \[\frac{1}{5} \|V\|_{\mathrm{BMO}^{p\mathrm{-var}},[0,T]} \leq \|V\|_{p,q,\infty,[0,T]} + \|\Gamma\|_{p,\infty,[0,T]} \leq C \|V\|_{\mathrm{BMO}^{p\mathrm{-var}},[0,T]}.\] In particular, \(V \in \mathrm{BMO}^{p\mathrm{-var}}\) if and only if \(V \in V^p L^{q,\infty}\) and \(\|\Gamma\|_{p,\infty,[0,T]} < \infty\).
Proof. For some \((s,t) \in \Delta_{[0,T]}\), we let \(A_u := \sup_{r \in [s,u]} |\delta V_{s,r}|\) for \(u \in [s,t]\). Then, by [45], for any stopping time \(s \leq \sigma \leq t\), we have that \[\big\|\mathbb{E}_\sigma [|\delta A_{\sigma-,t}|]\big\|_{L^\infty} \leq \rho_{s,t}(A) \leq 11 \rho_{s,t}(V).\] Hence, by [45], we have that \[\|\delta V_{s,t}\|_{q,\infty,s}^{\lceil q \rceil} \leq \big\|\mathbb{E}_s \big[|A_t|^{\lceil q \rceil}\big]\big\|_{L^\infty} \lesssim \rho_{s,t}(V)^{\lceil q \rceil},\] which implies that \(\|V\|_{p,q,\infty,[0,T]} \lesssim \|V\|_{\mathrm{BMO}^{p\text{-var}},[0,T]}\). By [45], we also have that \(\|\Gamma_{s,t}\|_{L^\infty} \leq \rho_{s,t}(V)\), so that \(\|\Gamma\|_{p,\infty, [0,T]} \leq \|V\|_{\mathrm{BMO}^{p\text{-var}},[0,T]}\).
For the reverse inequality, we apply [45] to find that \[\label{eq:32bound32on32BMO-p-var} \rho_{s,t}(V) \leq 2 \|V\|_{p,q,\infty,[s,t]} + 5 \|\Gamma_{s,t}\|_{L^\infty},\tag{24}\] so that \(\|V\|_{\mathrm{BMO}^{p\text{-var}},[0,T]} \leq 5 (\|V\|_{p,q,\infty,[0,T]} + \|\Gamma\|_{p,\infty,[0,T]})\). ◻
The following corollary is an immediate consequence of Proposition 12 and Theorem 2.
Corollary 1. Let \(p \in [1,\infty)\). If \(V \in \mathrm{BMO}^{p\mathrm{-var}}\) and \(V_0 \in L^\infty\), then \(V \in V^p L^{q,\infty}\) for every \(q \in [1,\infty)\). In particular, for any \(\tilde{p}> p\), we have that \[\big\| \|V\|_{\tilde{p},[0,T]} \big\|_{L^q} < \infty\] for every \(q \in [1,\infty)\).
Definition 13. We call a control \(w\) on \([0,T]\) regular from the inside if it is continuous from the inside, in the sense that \(w(s+,t) = w(s,t) = w(s,t-)\) for all \((s,t) \in \Delta_{[0,T]}\), and also satisfies \[\lim_{t \searrow s} w(s,t) = 0\] for every \(s \in [0,T)\).
The following definition and subsequent lemma may be found in [57], which itself builds on the results of [53]. Although here we consider slightly more general controls, the proof of Lemma 6 follows the corresponding proof in [57] verbatim.
Definition 14. Let \(w\) be a control which is regular from the inside. For a given \(\alpha > 0\) and \((s,t) \in \Delta_{[0,T]}\), we write \(t_0(\alpha) := s\), and \[\label{eq:32definition32tau95i40alpha41} t_i(\alpha) := \inf \{u > t_{i-1}(\alpha) \;| \;w(t_{i-1}(\alpha),u) \geq \alpha\} \wedge t\tag{25}\] for \(i \geq 1\), and then define \[N_{\alpha, [s,t]}(w) := \sup \{n \in \mathbb{N}\cup \{0\} \;| \;t_n(\alpha) < t\}.\]
Lemma 6 (Lemma 2 in [57]). Let \(w_1, w_2\) be controls which are regular from the inside, and let \(\alpha > 0\) and \((s,t) \in \Delta_{[0,T]}\). Suppose that \(w_1(u,v) \leq C w_2(u,v)\) holds for all \((u,v) \in \Delta_{[0,T]}\) such that \(w_2(u,v) \leq \alpha\), for some constant \(C > 0\). Then \[N_{C\alpha, [s,t]}(w_1) \leq N_{\alpha, [s,t]}(w_2).\]
The following result is a generalisation of [57], where here we need to take care of the large jumps separately.
Lemma 7. Let \(w\) be a control which is regular from the inside, and let \(0 < \alpha \leq \beta\). Then \[N_{\alpha, [s,t]}(w) \leq C \big(N_{\beta, [s,t]}(w) + 1\big)\] holds for any \((s,t) \in \Delta_{[0,T]}\), where the constant \(C > 0\) depends only on \(\alpha\) and \(\beta\).
Proof. For notational simplicity, we will write \(\{t_i\}_{i=0}^{N_{\alpha, [s,t]}(w)}\) for the sequence of times in \([s,t)\) defined in 25 . For \((u,v) \in \Delta_{[0,T]}\), we write \[\Delta_{u,v} w := w(u,v+) - w(u,v),\] and we then set \[N^{\Delta}_{\frac{\alpha}{2}, [s,t]}(w) := \# \Big\{i \in \{0, \ldots, N_{\alpha, [s,t]}(w) - 1\} \;: \Delta_{t_i,t_{i+1}} w > \frac{\alpha}{2} \Big\}.\] (Here it is crucial that we consider the times \(t_i = t_i(\alpha)\), rather than \(t_i(\frac{\alpha}{2})\).) We also define \[w_{\alpha}(s,t) := \sup_{\substack{\mathcal{P}= (u_i)_{i=1}^k \subset [s,t]\\ w(u_i,u_{i+1}) \leq \alpha \, \forall i}} \, \sum_{i=1}^{k-1} w(u_i,u_{i+1}).\]
We note that, for each \(i\), either \(w(t_i,t_{i+1}) = \alpha\) or \(\Delta_{t_i,t_{i+1}} w > 0\). Thus, \[\begin{align} \alpha N_{\alpha,[s,t]}(w) &\leq \sum_{i=0}^{N_{\alpha,[s,t]}(w)-1} w(t_i,t_{i+1}) + \alpha N^{\Delta}_{\frac{\alpha}{2},[s,t]}(w) + \sum_{i \, : \, \Delta_{t_i,t_{i+1}} w \leq \frac{\alpha}{2}} \Delta_{t_i,t_{i+1}} w\\ &\leq w_{\alpha}(s,t) + \alpha N^{\Delta}_{\frac{\alpha}{2},[s,t]}(w) + \frac{\alpha}{2} N_{\alpha, [s,t]}(w), \end{align}\] so that, rearranging, we obtain \[N_{\alpha,[s,t]}(w) \leq \frac{2}{\alpha} \big(w_{\alpha}(s,t) + \alpha N^{\Delta}_{\frac{\alpha}{2}, [s,t]}(w)\big) \leq \frac{2}{\alpha} \big(w_{\beta}(s,t) + \alpha N^{\Delta}_{\frac{\alpha}{2}, [s,t]}(w)\big).\] By the same proof as that of [53], one can show that \[w_{\beta}(s,t) \leq \beta \big(2 N_{\beta, [s,t]}(w) + 1\big),\] and it thus suffices to show that there exists a constant \(\tilde{C} > 0\), depending only on \(\alpha\) and \(\beta\), such that \[\label{eq:32N32Delta32alpha47232bounded32by32N32beta} N_{\frac{\alpha}{2}, [s,t]}^{\Delta}(w) \leq \tilde{C} N_{\beta, [s,t]}(w).\tag{26}\] We will show that this holds with \(\tilde{C} = \lceil \frac{2\beta}{\alpha} \rceil + 2\). To this end, we write \[(t^{\Delta}_j(\alpha))_{j=0}^{N^{\Delta}_{\frac{\alpha}{2},[s,t]}(w)} := \Big\{ t_j(\alpha) \;: \;\Delta_{t_j(\alpha),t_{j+1}(\alpha)} w > \frac{\alpha}{2} \Big\}\] for the times when \(w\) exceeds \(\alpha\) with a jump larger than \(\frac{\alpha}{2}\). For any \(0 \leq i \leq N_{\beta, [s,t]}(w) - 1\), we then have that \[\# \Big\{j \;: \;t^{\Delta}_j(\alpha) \in [t_i(\beta),t_{i+1}(\beta)] \Big\} \leq \bigg\lceil \frac{2\beta}{\alpha} \bigg\rceil + 2,\] since, otherwise, by the superadditivity of \(w(\cdot, \cdot+)\), we would have that \[\beta < \bigg(\bigg\lceil \frac{2\beta}{\alpha} \bigg\rceil + 1\bigg) \frac{\alpha}{2} \leq \sum_{t_i(\beta) \leq t^{\Delta}_j(\alpha) < t^{\Delta}_{j+1}(\alpha) < t_{i+1}(\beta)} \Delta_{t^{\Delta}_j(\alpha), t^{\Delta}_{j+1}(\alpha)} w \leq w(t_i(\beta), t_{i+1}(\beta)) \leq \beta.\] Since there are \(N_{\beta, [s,t]}(w)\) intervals of the form \([t_i(\beta),t_{i+1}(\beta)]\), the bound in 26 follows. ◻
Lemma 8. Let \(w_1, w_2\) be controls which are regular from the inside, and let \(\alpha > 0\) and \((s,t) \in \Delta_{[0,T]}\). Then \[N_{\alpha, [s,t]}(w_1 + w_2) \leq N_{\frac{\alpha}{2}, [s,t]}(w_1) + N_{\frac{\alpha}{2}, [s,t]}(w_2).\]
Proof. This is simply a consequence of the definition of \(N_{\alpha, [s,t]}\), and the fact that, if \(w_1(t_i,u) + w_2(t_i,u) \geq \alpha\), then either \(w_1(t_i,u) \geq \frac{\alpha}{2}\) or \(w_2(t_i,u) \geq \frac{\alpha}{2}\). ◻
Theorem 4 (John–Nirenberg inequality). Let \(V = (V_t)_{t \in [0,T]}\) be a \(\mathrm{BMO}^{p\mathrm{-var}}\) process for some \(p \in [1,\infty)\). Then, for any \(\alpha > 0\), any \(\lambda > 0\) and any \(r \in [0,T]\), we have that \[\Big\| \mathbb{E}_r \Big[ \exp \Big( \lambda \sup_{t \in [r,T]} |\delta V_{r,t}| \Big) \Big] \Big\|_{L^\infty} \leq \exp \Big( C \Big(N_{\alpha,[r,T]} \big( \|V\|_{\mathrm{BMO}^{p\mathrm{-var}},(\cdot,\cdot)}^p \big) + 1 \Big) \Big(1 + \sup_{t \in (r,T]} \|\Delta V_t\|_{L^\infty}\Big) \Big).\] where the constant \(C\) depends only on \(p, \alpha\) and \(\lambda\).
Theorem 4 follows from the following theorem, combined with Proposition 12 applied with \(q = 1\) to the process \((\delta V_{0,t})_{t \in [0,T]}\).
Theorem 5. Let \(p, q \in [1,\infty)\), and let \(V \in V^p L^{q,\infty}\) such that \(\|\Gamma\|_{p,\infty,[0,T]} < \infty\), where \(\Gamma\) is the two-parameter process defined in ?? . Let \(\alpha > 0\) and \(\lambda > 0\). Then there exists a constant \(C\), which depends only on \(p, \alpha\) and \(\lambda\), such that, for every \(r \in [0,T]\), we have that \[\label{eq:32JN32bound32for32VpLqinfty} \begin{align} \Big\| &\mathbb{E}_r \Big[ \exp \Big(\lambda \sup_{t \in [r,T]} |\delta V_{r,t}|\Big) \Big] \Big\|_{L^\infty}\\ &\leq \exp \Big( C \Big(N_{\alpha,[r,T]} \big(\|V\|_{p,1,\infty, [\cdot,\cdot)}^p + \|\Gamma\|_{p,\infty,(\cdot,\cdot)}^p\big) + 1 \Big) \Big(1 + \sup_{t \in (r,T]} \|\Delta V_t\|_{L^\infty}\Big) \Big). \end{align}\tag{27}\]
Proof. By [58], whenever \(\lambda \rho_{s,t}(V) \leq e^{-3}\), we have that \[\Big\| \mathbb{E}_s \Big[ \exp \Big( \lambda \sup_{u \in [s,t]} |\delta V_{s,u}| \Big) \Big] \Big\|_{L^\infty} \leq M,\] where \(M := 1 + \sum_{m=1}^{\infty} \frac{(c_m m)^m}{m!} e^{-3m} < \infty\) and \(c_m^m := m(1+\frac{1}{m})^{(m+1)^2}\).
With the convention that \(\|\Gamma\|_{p,\infty,(s,s)} := 0\) for all \(s \in [0,T]\), we have that \[\bar{w}(s,t) := \|V\|^p_{p,1,\infty,[s,t)} + \|\Gamma\|_{p,\infty,(s,t)}^p\] defines a control \(\bar{w}\) which is regular from the inside. We let \(\gamma = (10 \lambda e^3)^{-p}\), and consider the partition \(\{t_i(\gamma)\}_{i=0}^{N_{\gamma,[r,t]}(\bar{w})+1}\) of the interval \([r,t]\) as defined in 25 . For notational simplicity, we will write \(t_i = t_i(\gamma)\) and \(N = N_{\gamma,[r,T]}(\bar{w})\).
We define a process \(\widetilde{V}^i = (\widetilde{V}^i_t)_{t \in [t_i,t_{i+1}]}\) such that \(\widetilde{V}^i_t = V_t\) for \(t \in [t_i,t_{i+1})\), and \(\widetilde{V}^i_{t_{i+1}} = V_{t_{i+1}-}\). Then \(\|\widetilde{V}^i\|_{p,1,\infty,[t_i,t_{i+1}]} = \|V\|_{p,1,\infty,[t_i,t_{i+1})}\) and, setting \(\Delta \widetilde{V}^i_{t_i} := 0\), \[\Big\|\sup_{u \in [t_i,t_{i+1}]} |\Delta \widetilde{V}^i_u| \Big\|_{L^\infty} = \Big\|\sup_{u \in (t_i,t_{i+1})} |\Delta V_u| \Big\|_{L^\infty} \leq \|\Gamma\|_{p,\infty,(t_i,t_{i+1})}.\] Using 24 , and the fact that \(\bar{w}(t_i,t_{t+1})^{\frac{1}{p}} \leq \gamma^{\frac{1}{p}} = (10 \lambda e^3)^{-1}\), we then have that \[\lambda \rho_{t_i,t_{i+1}}(\widetilde{V}^{i}) \leq 5 \lambda \big(\|V\|_{p,1,\infty,[t_i,t_{i+1})} + \|\Gamma\|_{p,\infty,(t_i,t_{i+1})}\big) \leq 10 \lambda \bar{w}(t_i,t_{t+1})^{\frac{1}{p}} \leq e^{-3}.\] For any \(r \in [0,T)\), there exists an \(i_0 \in \{0, 1, \ldots, N\}\) such that \(r \in [t_{i_0},t_{i_{0}+1})\). Then \[\sup_{t \in [r,T]} |\delta V_{r,t}| \leq \sup_{t \in [r,t_{i_0+1}]} |\delta \widetilde{V}^{i_0}_{r,t}| + \sum_{i=i_0+1}^N \sup_{t \in [t_i,t_{i+1}]} |\delta \widetilde{V}^i_{t_i,t}| + \sum_{i=i_{0}+1}^{N+1} |\Delta V_{t_i}|,\] so that \[\begin{align} &\Big\| \mathbb{E}_r \Big[ \exp \Big( \lambda \sup_{t \in [r,T]} |\delta V_{r,t}| \Big) \Big] \Big\|_{L^\infty}\\ &\leq \bigg\| \mathbb{E}_r \bigg[ \exp \bigg( \lambda \bigg( \sup_{t \in [r,t_{i_0+1}]} |\delta \widetilde{V}^{i_0}_{r,t}| + \sum_{i=i_0+1}^N \sup_{t \in [t_i,t_{i+1}]} |\delta \widetilde{V}^i_{t_i,t}| + \sum_{i=i_{0}+1}^{N+1} |\Delta V_{t_i}| \bigg) \bigg) \bigg] \bigg\|_{L^\infty}\\ &\leq \Big\| \mathbb{E}_r \Big[ \exp \Big( \lambda \sup_{t \in [r,t_{i_0+1}]} |\delta \widetilde{V}^{i_0}_{r,t}| \Big) \Big] \Big\|_{L^\infty} \prod_{i=i_0+1}^N \Big\| \mathbb{E}_{t_i} \Big[ \exp \Big( \lambda \sup_{t \in [t_i,t_{i+1}]} |\delta \widetilde{V}^i_{t_i,t}| \Big) \Big] \Big\|_{L^\infty}\\ &\quad \times \exp \bigg( \lambda \sum_{i=i_0+1}^{N+1} \|\Delta V_{t_i}\|_{L^\infty} \bigg)\\ &\leq M^{N+1} \exp \Big(\lambda (N + 1) \sup_{t \in (r,T]} \|\Delta V_t\|_{L^\infty}\Big), \end{align}\] which implies that 27 holds with \(\alpha = \gamma\), and hence also for general \(\alpha\) by Lemma 7. ◻
In particular, the John–Nirenberg inequality implies the following result, the proof of which is given in Appendix 6.1.
Proposition 15. Let \(p \in [1,\infty)\). If \(V \in \mathrm{BMO}^{p\mathrm{-var}}\) is a one-dimensional process with \(V_0 \in L^\infty\), then \(\exp(V) \in V^p L^{q,r}\) for any \(q \in [1,\infty)\) and \(r \in [q,\infty)\).
Moreover, if \(1 \leq \tilde{q} \leq r < q < \infty\), and if \(\widetilde{V}\in \mathrm{BMO}^{p\mathrm{-var}}\) is another one-dimensional process with \(\widetilde{V}_0 \in L^\infty\), then we have the estimate \[\label{eq:32exp40V4132-32exp40tV4132estimate} \big\|\exp(V) - \exp(\widetilde{V})\big\|_{p,\tilde{q},r,[0,T]} \leq C \big(\|V_0 - \widetilde{V}_0\|_{L^q} + \|V - \widetilde{V}\|_{p,q,\frac{qr}{\tilde{q}},[0,T]}\big),\qquad{(2)}\] where the constant \(C > 0\) depends only on \(p, q, \tilde{q}, r, \|V_0\|_{L^\infty}, \|\widetilde{V}_0\|_{L^\infty}, \|V\|_{\mathrm{BMO}^{p\mathrm{-var}},[0,T]}\) and on \(\|\widetilde{V}\|_{\mathrm{BMO}^{p\mathrm{-var}},[0,T]}\).
Proposition 16. Let \(p \in [2,3)\), \(q \in [2,\infty)\), \(\mathbf{X}\in \mathscr{V}^p\), and let \((Y,Y') \in \mathcal{V}^{p,q,\infty}_X\) such that \(\|\mathbf{X}\|_{p,[0,T]}\), \(\sup_{s \in [0,T]} \|Y_s\|_{L^\infty}\), \(\sup_{s \in [0,T]} \|Y'_s\|_{L^\infty}\), \(\|Y\|_{p,q,\infty,[0,T]}\), \(\|Y'\|_{p,q,\infty,[0,T]}\) and \(\|\mathbb{E}_{\boldsymbol{\cdot}} R^Y\|_{\frac{p}{2},\infty,[0,T]}\) are all bounded by a constant \(L > 0\). Then the rough stochastic integral \(\int_0^\cdot Y_s \,\mathrm{d}\mathbf{X}_s \in \mathrm{BMO}^{p\mathrm{-var}}\). Moreover, for any \(\alpha > 0\) and \(\lambda > 0\), there exists a constant \(C\), which depends only on \(p, q, \alpha, \lambda\) and \(L\), such that, for every \(r \in [0,T]\), we have that \[\label{eq:32exponential32bound32for32rough32stochastic32integral} \bigg\| \mathbb{E}_r \bigg[ \exp \bigg( \lambda \sup_{t \in [r,T]} \bigg| \int_r^t Y_s \,\mathrm{d}\mathbf{X}_s \bigg| \bigg) \bigg] \bigg\|_{L^\infty} \leq \exp \Big( C \Big(N_{\alpha,[r,T]}\big(\|\mathbf{X}\|_{p,[\cdot,\cdot)}^p\big) + 1 \Big) \Big(1 + \sup_{t \in (r,T]} |\Delta \mathbf{X}_t| \Big) \Big).\qquad{(3)}\]
Proof. For each \((s,t) \in \Delta_{[0,T]}\), we let \[\Gamma_{s,t} := \sup_{r \in [s,t]} \bigg|\Delta \bigg(\int_0^\cdot Y_u \,\mathrm{d}\mathbf{X}_u\bigg)_{r}\bigg| = \sup_{r \in [s,t]} \big|Y_{r-} \Delta X_r + Y'_{r-} \Delta \mathbb{X}_r\big|,\] where the second equality follows from the jump structure of rough stochastic integrals; see [22]. Let us write \(J_\mathbf{X}\subset (0,T]\) for the (countable) set of jump times of \(\mathbf{X}= (X,\mathbb{X})\). Then, for any partition \(\mathcal{P}\), we have that \[\begin{align} \sum_{[u,v] \in \mathcal{P}} \|\Gamma_{u,v}\|_{L^\infty}^p &= \sum_{[u,v] \in \mathcal{P}} \sup_{r \in [u,v] \cap J_\mathbf{X}} \big\|Y_{r-} \Delta X_r + Y'_{r-} \Delta \mathbb{X}_r\big\|_{L^\infty}^p\\ &\leq 2 \sum_{r \in J_\mathbf{X}} \big(\|Y_{r-}\|_{L^\infty} + \|Y'_{r-}\|_{L^\infty}\big)^p |\Delta \mathbf{X}_r|^p\\ &\lesssim \sup_{u \in [0,T]} \big(\|Y_u\|_{L^\infty} + \|Y'_u\|_{L^\infty}\big)^p \|\mathbf{X}\|_{p,[0,T]}^p \leq 2^p L^p \|\mathbf{X}\|_{p,[0,T]}^p. \end{align}\] It follows that \(\|\Gamma\|_{p,\infty,(s,t)} \lesssim \|\mathbf{X}\|_{p,[s,t)}\) for any \((s,t) \in \Delta_{[0,T]}\), and it is also straightforward to see that \[\sup_{t \in (0,T]} \bigg\| \Delta \bigg(\int_0^\cdot Y_u \,\mathrm{d}\mathbf{X}_u\bigg)_{t} \bigg\|_{L^\infty} \lesssim \sup_{t \in (0,T]} |\Delta \mathbf{X}_t|.\] By [22], we also have that \[\bigg\| \int_0^\cdot Y_u \,\mathrm{d}\mathbf{X}_u \bigg\|_{p,q,\infty,[s,t)} \lesssim \|\mathbf{X}\|_{p,[s,t)}\] for any \((s,t) \in \Delta_{[0,T]}\). It thus follows from Proposition 12 that \(\int_0^\cdot Y_u \,\mathrm{d}\mathbf{X}_u \in \mathrm{BMO}^{p\mathrm{-var}}\), and the bound in ?? follows from Theorem 5, combined with Lemma 6 and Lemma 7. ◻
We generalize the notion of locally linear maps in [57] to the setting of rough stochastic analysis. Taking the Itô–Lyons map as prime example, the presence of additional noise terms, not present in classical RDEs, necessitates a slightly different definition.
Definition 17. Let \(p \in [2,3)\) and \(q \in [2, \infty)\). We call a map \(\Psi \colon \mathscr{V}^p \to V^p L^{q,\infty}\) locally affine if there exists an \(R \in (0,\infty]\), and a control \(\bar{w}\) which is regular from the inside (in the sense of Definition 13), such that \(\|\Psi\|_{R,\bar{w}} < \infty\), where \[\begin{align} \|\Psi\|_{R,\bar{w}} := \inf \big\{ C > 0 \, : \;&\|\Psi(\mathbf{X})\|_{p,q,\infty,[s,t)}^p \leq C \bar{w}_{\mathbf{X}}(s,t)\\ &\text{for all } \mathbf{X}\in \mathscr{V}^p \text{ and } (s,t) \in \Delta_{[0,T]} \text{ such that } \bar{w}_{\mathbf{X}}(s,t) \leq R \big\} \end{align}\] and \(\bar{w}_{\mathbf{X}}(s,t) := \|\mathbf{X}\|_{p,[s,t)}^p + \bar{w}(s,t)\).
With this definition, we easily conclude an analogue of [57] in our context. The following result follows as a direct consequence of Lemmas 6 and 8.
Lemma 9. Let \(\Psi \colon \mathscr{V}^p \to V^p L^{q,\infty}\) be a locally affine map, for some \(R \in (0,\infty]\) and control \(\bar{w}\). Then \[N_{\alpha \|\Psi\|_{R,\bar{w}},[s,t]}\big(\|\Psi(\mathbf{X})\|_{p,q,\infty,[\cdot,\cdot)}^p\big) \leq N_{\frac{\alpha}{2},[s,t]}\big(\|\mathbf{X}\|_{p,[\cdot,\cdot)}^p\big) + N_{\frac{\alpha}{2},[s,t]}(\bar{w})\] for every \(\alpha \in (0,R]\).
Proposition 18. Under the assumptions of Theorem 1, the Itô–Lyons map \[\mathscr{V}^p \ni \mathbf{X}\mapsto Y \in V^p L^{q,\infty},\] where \(Y\) is the solution to the rough SDE 6 driven by \(\mathbf{X}\), is locally affine, with any \(R \in (0,\infty)\) and the control \(\bar{w}\) given by \[\bar{w}(s,t) := (t - s) + \|M\|_{p,q,\infty,[s,t)}^p + \|A\|_{\frac{p}{2},\frac{q}{2},\infty,[s,t)}^{\frac{p}{2}} + \|A\|_{\frac{p}{q},1,\infty,[s,t)}^{\frac{p}{q}}.\]
Moreover, for any \(h \in C^2_b\), the map \(\mathbf{X}\mapsto \int_0^{\cdot} h(Y_s) \,\mathrm{d}\mathbf{X}_s\) is also locally affine, with any \(R \in (0,\infty)\) and the same control \(\bar{w}\). In particular, if \(\mathbf{X}\in \mathscr{V}^p\) with \(\|\mathbf{X}\|_{p,[0,T]}, \bar{w}(0,T) \leq L\) for some \(L > 0\), then for any \(\alpha > 0\), \(\lambda > 0\) and \(r \in [0,T]\), \[\label{eq:32exponential32bound32for32rough32stochastic32integral32of32RSDE32solutions} \bigg\| \mathbb{E}_r \bigg[ \exp \bigg( \lambda \sup_{t \in [r,T]} \bigg| \int_r^t h(Y_s) \,\mathrm{d}\mathbf{X}_s \bigg| \bigg) \bigg] \bigg\|_{L^\infty} \leq \exp \Big( C \Big(N_{\alpha,[r,T]}\big(\|\mathbf{X}\|_{p,[\cdot,\cdot)}^p\big) + 1 \Big) \Big(1 + \sup_{t \in (r,T]} |\Delta \mathbf{X}_t| \Big) \Big),\qquad{(4)}\] where the constant \(C\) depends only on \(p, q, \alpha, \lambda\) and \(L\).
Proof. By Step 1 in the proof of [22] (suitably adapted to include the integral against a random measure), there exists an \(\varepsilon\in (0,1]\), which does not depend on \(\mathbf{X}\), such that, for every \((s,t) \in \Delta_{[0,T]}\) with \(\bar{w}_\mathbf{X}(s,t) \leq \varepsilon\), we have that \[\label{eq:32bound32for32Y32and32ER94Y32in32loc32affine32proof} \|Y\|_{p,q,\infty,[s,t)}^p \vee \|\mathbb{E}_{\boldsymbol{\cdot}} R^Y\|_{\frac{p}{2},\infty,[s,t)}^{\frac{p}{2}} \lesssim \bar{w}_\mathbf{X}(s,t).\tag{28}\] Since \(Y' = f(Y)\) and \(f\) is Lipschitz, we also have that \(\|Y'\|_{p,q,\infty,[s,t)}^p \lesssim \|Y\|_{p,q,\infty,[s,t)}^p \lesssim \bar{w}_\mathbf{X}(s,t)\).
We now fix any \(\mathbf{X}\in \mathscr{V}^p\) and \((s,t) \in \Delta_{[0,T]}\) such that \(w_{\mathbf{X}}(s,t) \leq R\). By [46], there exists a partition \(\{t_i\}_{i=0}^M\) of the interval \([s,t]\) such that \(w_{\mathbf{X}}(t_i,t_{i+1}) \leq \varepsilon\) for each \(0 \leq i < M\). By the superadditivity of \(\bar{w}_\mathbf{X}\), we may also choose this partition such that \(M \lesssim \bar{w}_\mathbf{X}(s,t) \leq R\).
For \(0 \leq i \leq M-2\), using [22], and the fact that \[\|f(Y)\|_{p,q,\infty,[v,t_{i+1})} + \|f(Y)'\|_{p,q,\infty,[v,t_{i+1})} + \|\mathbb{E}_{\boldsymbol{\cdot}} R^{f(Y)}\|_{\frac{p}{2},\infty,[v,t_{i+1})} \to 0\] as \(v \nearrow t_{i+1}\), we see that \[\lim_{v \nearrow t_{i+1}} \bigg\| \int_v^{t_{i+1}} f(Y_r) \,\mathrm{d}\mathbf{X}_r \bigg\|_{q,\infty,v}^p \lesssim \|\mathbf{X}\|_{p,[t_i,t_{i+1}]}^p \leq \|\mathbf{X}\|_{p,[s,t)}^p.\] Using the bound in 28 , the conditional BDG inequality and Lemma 16, we then have that \[\begin{align} \|&Y\|_{p,q,\infty,[t_i,t_{i+1}]}^p \lesssim \|Y\|_{p,q,\infty,[t_i,t_{i+1})}^p + \lim_{v \nearrow t_{i+1}} \|\delta Y_{v,t_{i+1}}\|_{q,\infty,v}^p\\ &\lesssim \bar{w}_\mathbf{X}(t_i,t_{i+1}) + \lim_{v \nearrow t_{i+1}} \bigg\| \int_v^{t_{i+1}} \sigma(Y_{r-}) \,\mathrm{d}M_r \bigg\|_{q,\infty,v}^p + \lim_{v \nearrow t_{i+1}} \bigg\| \int_v^{t_{i+1}} f(Y_r) \,\mathrm{d}\mathbf{X}_r \bigg\|_{q,\infty,v}^p\\ &\quad + \lim_{v \nearrow t_{i+1}} \bigg\| \int_v^{t_{i+1}} \int_{\mathbb{U}} g(r,Y_{r-},u) \, \widetilde{N}(\mathrm{d}u,\mathrm{d}r) \bigg\|_{q,\infty,v}^p\\ &\lesssim \bar{w}_\mathbf{X}(t_i,t_{i+1}) + \|M\|_{p,q,\infty,[t_i,t_{i+1}]}^p + \|A\|_{\frac{p}{2},\frac{q}{2},\infty,[t_i,t_{i+1}]}^{\frac{p}{2}} + \|A\|_{\frac{p}{q},1,\infty,[t_i,t_{i+1}]}^{\frac{p}{q}} + \|\mathbf{X}\|_{p,[s,t)}^p\\ &\lesssim \bar{w}_\mathbf{X}(s,t). \end{align}\] It is also clear that \(\|Y\|_{p,q,\infty,[t_{M-1},t_M)}^p \lesssim \bar{w}_\mathbf{X}(t_{M-1},t_M) \leq \bar{w}_\mathbf{X}(s,t)\), and we can thus bound \[\|Y\|_{p,q,\infty,[s,t)}^p \leq M^{p-1} \bigg( \sum_{i=0}^{M-2} \|Y\|_{p,q,\infty,[t_i,t_{i+1}]}^p + \|Y\|_{p,q,\infty,[t_{M-1},t_M)}^p \bigg) \lesssim \bar{w}_\mathbf{X}(s,t),\] which implies that the map \(\mathbf{X}\mapsto Y\) is locally affine.
With the same line of argument, one can conclude similar bounds for \(\|Y'\|_{p,q,\infty,[s,t)}^p\) and \(\|\mathbb{E}_{\boldsymbol{\cdot}} R^Y\|_{\frac{p}{2},\infty,[s,t)}^{\frac{p}{2}}\), and the final claim then follows by combining these bounds with those in [22], and arguing exactly as in the proof of Proposition 16. ◻
On a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t \in [0,T]},\mathbb{P})\), we let \(B\) and \(W\) be independent Brownian motions. Moreover, we let \(N_1\) and \(N_2\) be integer-valued random measures, defined on Blackwell spaces \((\mathbb{U}_1,\mathcal{U}_1)\), \((\mathbb{U}_2,\mathcal{U}_2)\), and let \(\widetilde{N}_1\), \(\widetilde{N}_2\) denote the corresponding compensated random measures.
We consider a signal process \(X\), and an observation process \(Y\), governed by the SDEs \[\label{eq:32SDEofX32and32Y} \begin{align} \mathrm{d}X_t &= b_1(t,X_t,Y_t) \,\mathrm{d}t + \sigma_0(t,X_t,Y_t) \,\mathrm{d}B_t + \sigma_1(t,X_t,Y_t) \,\mathrm{d}W_t\\ &\quad + \int_{\mathbb{U}_1} f_1(t,X_{t-},Y_{t-},u) \, \widetilde{N}_1(\mathrm{d}t,\mathrm{d}u) + \int_{\mathbb{U}_2} f_2(t,X_{t-},Y_{t-},u) \, \widetilde{N}_2(\mathrm{d}t,\mathrm{d}u),\\ \mathrm{d}Y_t &= b_2(t,X_t,Y_t) \,\mathrm{d}t + \sigma_2(t,Y_t) \,\mathrm{d}W_t + \int_{\mathbb{U}_2} f_3(t,Y_{t-},u) \, \widetilde{N}_2(\mathrm{d}t,\mathrm{d}u). \end{align}\tag{29}\] We suppose that compensators of \(N_1\) and \(N_2\) are given by \(\nu_1(\mathrm{d}u) \,\mathrm{d}t\) and \(\lambda(t,X_{t-},u) \, \nu_2(\mathrm{d}u) \,\mathrm{d}t\) respectively, where \(\nu_1\), \(\nu_2\) are \(\sigma\)-finite measures on the respective Blackwell spaces, and \(\lambda \colon [0,T] \times \mathbb{R}^{d_X} \times \mathbb{U}_2 \to (0,\infty)\) is a Borel measurable function. In particular, we note that \(N_1\) is a Poisson random measure, but \(N_2\) is not in general.
Here, the coefficients \(b_1 \colon [0,T] \times \mathbb{R}^{d_X + d_Y} \to \mathbb{R}^{d_X}\), \(b_2 \colon [0,T] \times \mathbb{R}^{d_X + d_Y} \to \mathbb{R}^{d_Y}\), \(\sigma_0 \colon [0,T] \times \mathbb{R}^{d_X + d_Y} \to \mathbb{R}^{d_X \times d_B}\), \(\sigma_1 \colon [0,T] \times \mathbb{R}^{d_X + d_Y} \to \mathbb{R}^{d_X \times d_Y}\), \(\sigma_2 \colon [0,T] \times \mathbb{R}^{d_Y} \to \mathbb{R}^{d_Y \times d_Y}\), \(f_1 \colon [0,T] \times \mathbb{R}^{d_X + d_Y} \times \mathbb{U}_1 \to \mathbb{R}^{d_X}\), \(f_2 \colon [0,T] \times \mathbb{R}^{d_X + d_Y} \times \mathbb{U}_2 \to \mathbb{R}^{d_X}\) and \(f_3 \colon [0,T] \times \mathbb{R}^{d_Y} \times \mathbb{U}_2 \to \mathbb{R}^{d_Y}\) are all Borel measurable, and \(d_X, d_Y, d_B \in \mathbb{N}\) are the dimensions of the respective processes. We note that this setting includes, for instance, those in [36] and [40].
Stochastic filtering is concerned with calculating the conditional law of \(X_t\) given the filtration generated by \(Y\) up to time \(t\). All the characteristics of \(X\) and \(Y\) are assumed to be known by the observer; see also Remark 27. Robust stochastic filtering establishes continuity properties of the conditional law with respect to the observation \(Y\), which is of course crucial in applications.
We require the following assumptions.
Assumption 2. The coefficients have at most linear growth. That is, there exists a constant \(K > 0\) such that \[\begin{align} |b_1(t,x,y)| + |b_2(t,x,y)| + |\sigma_0(t,x,y)| + |\sigma_1(t,x,y)| + |\sigma_2(t,y)| &\leq K (1 + |x| + |y|),\\ \|f_1(t,x,y,\cdot)\|_{L^2(\nu_1)} + \|f_2(t,x,y,\cdot)\|_{L^2(\nu_2)} + \|f_3(t,y,\cdot)\|_{L^2(\nu_2)} &\leq K(1 + |x| + |y|) \end{align}\] for all \(t \in [0,T]\), \(x \in \mathbb{R}^{d_X}\) and \(y \in \mathbb{R}^{d_Y}\).
Assumption 3. The initial values \(X_0\), \(Y_0\) are \(\mathcal{F}_0\)-measurable and square integrable.
Assumption 4. Almost surely, the random measures \(N_1\) and \(N_2\) do not jump at the same time, in the sense that \(\mathbb{P}(\{\omega \in \Omega : (\omega,t) \in D_1 \cap D_2 \;\mathrm{ for any } \;t \in [0,T]\}) = 0\), where \(D_i\) denotes the set of jump times of \(N_i\), as defined in 5 .
Assumption 5. The map \(\sigma_2 \colon [0,T] \times \mathbb{R}^{d_Y} \to \mathbb{R}^{d_Y \times d_Y}\) takes values in the space of invertible matrices, and \[\sup_{t \in [0,T], \, y \in \mathbb{R}^{d_Y}} |\sigma_2(t,y)^{-1}| < \infty.\] Further the map \(h \colon [0,T] \times \mathbb{R}^{d_X + d_Y} \to \mathbb{R}^{d_Y}\) given by \[\label{eq:32defn32function32h} h(t,x,y) := \sigma_2(t,y)^{-1} \bigg( b_2(t,x,y) + \int_{\mathbb{U}_2} f_3(t,y,u) (1 - \lambda(t,x,u)) \, \nu_2(\mathrm{d}u) \bigg)\tag{30}\] satisfies \[\sup_{(x,y) \in \mathbb{R}^{d_X + d_Y}} \int_0^T |h(s,x,y)|^2 \,\mathrm{d}s < \infty.\]
Assumption 6. The function \(\lambda\) is uniformly bounded and uniformly bounded away from \(0\). Further, \[\label{eq:32assumption32on32lambda} \sup_{t \in [0,T], \, x \in \mathbb{R}^{d_X}} \int_{\mathbb{U}_2} \frac{(1 - \lambda(t,x,u))^2}{\lambda(t,x,u)} \, \nu_2(\mathrm{d}u) < \infty.\tag{31}\]
Conditions of the form in 31 are standard for filtering models with Lévy noise; see, e.g., [59], [40], [41] or [31].
The following lemma provides some required integrability. Since this result is rather standard (see, e.g., [44] or [36]), we omit its proof.
Lemma 10. Suppose that \(X\) and \(Y\) satisfy the SDEs in 29 . Then, under Assumptions 2 and 3, we have that \[\mathbb{E}\Big[ \sup_{t \in [0,T]} \big(|X_t|^2 + |Y_t|^2\big) \Big] \leq C \big( 1 + \mathbb{E}\big[|X_0|^2\big] + \mathbb{E}\big[|Y_0|^2\big] \big) < \infty,\] where the constant \(C\) depends in particular on the constant \(K\) in Assumption 2, which implies in particular that \[\mathbb{E}\bigg[ \int_0^T \int_{\mathbb{U}_i} |f_i(s,X_{s-},Y_{s-},u)|^2 \, \nu_i(\mathrm{d}u) \,\mathrm{d}s \bigg] < \infty\] for each \(i = 1, 2, 3\) (with \(\mathbb{U}_3 := \mathbb{U}_2\), \(\nu_3 := \nu_2\) and without the dependence on \(X\) when \(i = 3\)).
A standard approach in stochastic filtering, which we will utilize here, is the so-called reference measure method. In the following, we will define a probability measure \(\widetilde{\mathbb{P}}\), which is equivalent to \(\mathbb{P}\) on \(\mathcal{F}_T\), under which the noises driving the observation process \(Y\) become independent of the additional noises driving the signal process \(X\). In our setting, this is achieved by setting \[\frac{\mathrm{d}\widetilde{\mathbb{P}}}{\mathrm{d}\mathbb{P}}\bigg|_{\mathcal{F}_T} = \Lambda_T^{-1},\] where \[\Lambda_t^{-1} := \exp(-I_t)\] for \(t \in [0,T]\), and the process \(I\) is defined by \[\label{eq:32defn32I95t} \begin{align} I_t &= \int_0^t h(s,X_s,Y_s)^\top \,\mathrm{d}W_s + \frac{1}{2} \int_0^t |h(s,X_s,Y_s)|^2 \,\mathrm{d}s + \int_0^t \int_{\mathbb{U}_2} \log(\lambda(s,X_{s-},u)) \, \widetilde{N}_2(\mathrm{d}s,\mathrm{d}u)\\ &\quad + \int_0^t \int_{\mathbb{U}_2} \big( 1 - \lambda(s,X_{s-},u) + \lambda(s,X_{s-},u) \log(\lambda(s,X_{s-},u)) \big) \, \nu_2(\mathrm{d}u) \,\mathrm{d}s \end{align}\tag{32}\] for \(t \in [0,T]\), and the function \(h\) was defined in 30 , so that \[h(s,X_s,Y_s) = \sigma_2(s,Y_s)^{-1} \bigg( b_2(s,X_s,Y_s) + \int_{\mathbb{U}_2} f_3(s,Y_{s-},u) (1 - \lambda(s,X_{s-},u)) \, \nu_2(\mathrm{d}u) \bigg).\]
Since \((v - 1)/v \leq \log(v) \leq v - 1\) for all \(v > 0\), and \(\lambda\) is uniformly bounded, we note that \[\big| 1 - \lambda(s,x,u) + \lambda(s,x,u) \log(\lambda(s,x,u)) \big| \leq (1 - \lambda(s,x,u))^2 \lesssim \frac{(1 - \lambda(s,x,u))^2}{\lambda(s,x,u)},\] which, combined with Assumption 6, ensures the existence of the final integral in 32 .
The following few lemmas are also standard, and their proofs are rather technical, and are therefore omitted. See, for instance, [59], [40], [1] or [60] for similar results in analogous settings.
Lemma 11. Under Assumptions 2, 3, 5 and 6, we have that \(\Lambda^{-1} = (\Lambda^{-1}_t)_{t \in [0,T]}\) is the Doléans–Dade exponential of the process given by \[t \mapsto -\int_0^t h(s,X_s,Y_s)^\top \,\mathrm{d}W_s + \int_0^t \int_{\mathbb{U}_2} \frac{1 - \lambda(s,X_{s-},u)}{\lambda(s,X_{s-},u)} \, \widetilde{N}_2(\mathrm{d}s,\mathrm{d}u),\] and thus in particular is an exponential martingale.
We let \(\widetilde{W}\) be the process given by \(\widetilde{W}_t = W_t + \int_0^t h(s,X_s,Y_s) \,\mathrm{d}s\) for \(t \in [0,T]\), and we also let \(\tilde{b}_1(t,x,y) := b_1(t,x,y) - \sigma_1(t,x,y) h(t,x,y) - \int_{\mathbb{U}_2} f_2(t,x,y,u) (1 - \lambda(t,x,u)) \, \nu_2(\mathrm{d}u)\).
Lemma 12. Under Assumptions 2–6, we have, under the measure \(\widetilde{\mathbb{P}}\), that \(B\) and \(\widetilde{W}\) are Brownian motions, \(N_1\) and \(N_2\) are Poisson random measures with compensators \(\nu_1(\mathrm{d}u) \,\mathrm{d}t\) and \(\nu_2(\mathrm{d}u) \,\mathrm{d}t\) respectively, and that \(B, \widetilde{W}, N_1\) and \(N_2\) are all independent.
Moreover, writing \(\widetilde{N}\) for the compensated random measure associated with \(N_2\) under \(\widetilde{\mathbb{P}}\), i.e., \(\widetilde{N}(\mathrm{d}t,\mathrm{d}u) = N_2(\mathrm{d}t,\mathrm{d}u) - \nu_2(\mathrm{d}u) \,\mathrm{d}t\), the signal \(X\) and observation \(Y\) satisfy \[\label{eq:32SDEFilteringUpdated} \begin{align} \mathrm{d}X_t &= \tilde{b}_1(t,X_t,Y_t) \,\mathrm{d}t + \sigma_0(t,X_t,Y_t) \,\mathrm{d}B_t + \sigma_1(t,X_t,Y_t) \,\mathrm{d}\widetilde{W}_t\\ &\quad + \int_{\mathbb{U}_1} f_1(t,X_{t-},Y_{t-},u) \, \widetilde{N}_1(\mathrm{d}t,\mathrm{d}u) + \int_{\mathbb{U}_2} f_2(t,X_{t-},Y_{t-},u) \, \widetilde{N}(\mathrm{d}t,\mathrm{d}u),\\ \mathrm{d}Y_t &= \sigma_2(t,Y_t) \,\mathrm{d}\widetilde{W}_t + \int_{\mathbb{U}_2} f_3(t,Y_{t-},u) \, \widetilde{N}(\mathrm{d}t,\mathrm{d}u). \end{align}\tag{33}\]
Under standard assumptions (see, e.g., [35] or [44]), there exists a unique solution \(X, Y\) to the SDEs in 33 , which is then also the solution to the SDEs in 29 .
The conditional distribution for our filtering model is given by the following result, known as the Kallianpur–Striebel formula. In the following, we will write \(\widetilde{\mathbb{E}}\) for the expectation under \(\widetilde{\mathbb{P}}\), and write \((\mathcal{F}^Y_t)_{t \in [0,T]}\) for the observation filtration, i.e., for each \(t \in [0,T]\), \(\mathcal{F}^Y_t\) is the \(\mathbb{P}\)-completion of \(\sigma(Y_s, s \in [0,t])\).
Lemma 13. For any bounded measurable function \(f \colon \mathbb{R}^{d_X + d_Y} \to \mathbb{R}\), we have that \[\pi_t(f) := \mathbb{E}[f(X_t,Y_t) \, | \, \mathcal{F}^Y_t] = \frac{\widetilde{\mathbb{E}}[f(X_t,Y_t) \Lambda_t \, | \, \mathcal{F}^Y_t]}{\widetilde{\mathbb{E}}[\Lambda_t \, | \, \mathcal{F}^Y_t]} =: \frac{\rho_t(f)}{\rho_t(1)}\] for \(t \in [0,T]\), where \(\Lambda_t = \exp(I_t)\). Moreover, \(\Lambda = (\Lambda_t)_{t \in [0,T]}\) is the Doléans–Dade exponential of the process \[t \mapsto \int_0^t h(s,X_s,Y_s)^\top \,\mathrm{d}\widetilde{W}_s - \int_0^t \int_{\mathbb{U}_2} (1 - \lambda(s,X_{s-},u)) \, \widetilde{N}(\mathrm{d}s,\mathrm{d}u),\] and thus is itself an exponential martingale.
Similarly to the approach adopted in [13], rather than working with the SDEs 33 on a common probability space, we consider analogous SDEs on a product probability space, such that the additional noise terms included in the signal are defined on an independent part of the space to the observation. By the independence of these terms, as provided in Lemma 12, the solution to the SDEs on the product space has the same distribution as the solution to the original SDEs in 33 . Moreover, we can then lift all the \(\mathcal{F}^Y_t\)-adapted processes to rough paths, and, given the consistency result of Section 3, we can consider the resulting equation as a rough SDE, and exploit the stability properties of such equations to obtain robustness of the conditional distribution. To this end, we need the following assumption.
Assumption 7. We assume that \[\mathcal{F}^Y_t = \mathcal{F}^Y_0 \vee \mathcal{F}^{\widetilde{W}}_t \vee \mathcal{F}^{\widetilde{N}}_t\] for every \(t \in [0,T]\). Here, \((\mathcal{F}^{\widetilde{W}}_t)_{t \in [0,T]}\) denotes the natural filtration of \(\widetilde{W}\), and \(\mathcal{F}^{\widetilde{N}}_t\) is the \(\widetilde{\mathbb{P}}\)-completion of the \(\sigma\)-algebra generated by the random variables \(N_2((0,s] \times U)\) for all \(s \in (0,t]\) and all \(U \in \mathcal{U}_2\) such that \(\nu_2(U) < \infty\). We also assume that \(\mathcal{F}^Y_0\) is independent of \(\mathcal{F}^B_T \vee \mathcal{F}^{\widetilde{N}_1}_T\), where \((\mathcal{F}^B_t)_{t \in [0,T]}\) is the natural filtration of \(B\), and \(\mathcal{F}^{\widetilde{N}_1}_t\) is defined analogously to \(\mathcal{F}^{\widetilde{N}}_t\).
Remark 19. Assumption 7 is essentially an additional condition on the coefficient \(f_3\).
Suppose, for instance, that \(f_3(s,y,u) = h_3(s,y) g_3(s,u)\), where \(h_3\) takes values in the space of invertible \(d_Y \times d_Y\)-matrices with \(\sup_{s \in [0,T], \, y \in \mathbb{R}^{d_Y}} |h_3(s,y)^{-1}| < \infty\), and \(g_3 \colon [0,T] \times \mathbb{U}_2 \to \mathbb{R}^{d_Y} \setminus \{0\}\) is such that \(\sigma(\iota|_{[0,t] \times \mathbb{U}_2}) = \mathcal{B}([0,t]) \otimes \mathcal{U}_2\) for every \(t \in [0,T]\), where the function \(\iota \colon [0,T] \times \mathbb{U}_2 \to [0,T] \times (\mathbb{R}^{d_Y} \setminus \{0\})\) is given by \(\iota(s,u) = (s,g_3(s,u))\). An extension of [36] then shows that Assumption 7 is satisfied.
As indicated above, our intention is to fix a realization of certain noise terms in 33 , and then lift these to a rough path. While this procedure is classical for Itô integrals against Brownian motion, the Poisson random measures cannot immediately be lifted to a rough path; indeed, a priori there is no path to lift. We circumvent this issue by splitting the integrand into a product, to isolate the dependence on \(X\) and \(Y\) from the dependence on \(u\).
Assumption 8. We suppose that \(\tilde{b}_1, \sigma_0 \in C^1_b\), \(h \in C^2_b\) and \(\sigma_1, \sigma_2 \in C^3_b\). Further, we assume that \(f_2(t,x,y,u) = h_2(t,x,y) g_2(t,u)\) and \(f_3(t,y,u) = h_3(t,y) g_3(t,u)\), where \(h_2, h_3 \in C^3_b\), and, for each \(i = 2, 3\), the function \(g_i\) is Borel measurable and satisfies \[\int_0^T \int_{\mathbb{U}_2} |g_i(s,u)|^2 \, \nu_2(\mathrm{d}u) \,\mathrm{d}s < \infty.\] Moreover, we assume that \(\log(\lambda(t,x,u)) = \kappa(t,x) \gamma(t,u)\), where \(\kappa \in C^2_b\) and \(\gamma\) is Borel measurable, bounded, and satisfies \[\int_0^T \int_{\mathbb{U}_2} |\gamma(s,u)| \, \nu_2(\mathrm{d}u) \,\mathrm{d}s < \infty.\]
Remark 20. The conditions on \(f_2, f_3\) and \(\lambda\) in Assumption 8 may seem somewhat restrictive at first glance. However, we recall that finite sums of products of smooth functions, of the form \(\sum_{j=1}^m h_{2,j}(t,x,y) g_{2,j}(t,u)\), are dense in the space of continuous compactly supported functions of \((t,x,y,u)\), by a straightforward application of the Stone–Weierstrass theorem. Indeed, all of our results below hold when the functions \(f_2, f_3\) and \(\lambda\) are given by finite sums of products, i.e., when \(f_2(t,x,y,u) = \sum_{j=1}^m h_{2,j}(t,x,y) g_{2,j}(t,u)\) (and similarly for \(f_3\)), and \(\log(\lambda(t,x,u)) = \sum_{j=1}^m \kappa_j(t,x) \gamma_j(t,u)\), where the functions \(h_{2,j}, h_{3,j}, g_{2,j}, g_{3,j}, \kappa_j\) and \(\gamma_j\) satisfy the relevant conditions in Assumption 8. It what follows we take singular products purely for notational simplicity.
Under Assumption 8, and using the associativity of integrals against random measures (e.g., [47]), we may rewrite 33 as \[\label{eq:32dynamics32of32X32Y32driven32by32G} \begin{align} \mathrm{d}X_t &= \tilde{b}_1(t,X_t,Y_t) \,\mathrm{d}t + \sigma_0(t,X_t,Y_t) \,\mathrm{d}B_t + \int_{\mathbb{U}_1} f_1(t,X_{t-},Y_{t-},u) \, \widetilde{N}_1(\mathrm{d}t,\mathrm{d}u)\\ &\quad + (\sigma_1(t,X_{t-},Y_{t-}), h_2(t,X_{t-},Y_{t-}), 0, 0) \,\mathrm{d}G_t,\\ \mathrm{d}Y_t &= (\sigma_2(t,Y_{t-}), 0, h_3(t,Y_{t-}), 0) \,\mathrm{d}G_t, \end{align}\tag{34}\] where we set \[\label{eq:32definition32G} G_t = \bigg( \widetilde{W}_t, \int_0^t \int_{\mathbb{U}_2} g_2(s,u) \, \widetilde{N}(\mathrm{d}s,\mathrm{d}u), \int_0^t \int_{\mathbb{U}_2} g_3(s,u) \, \widetilde{N}(\mathrm{d}s,\mathrm{d}u), \int_0^t \int_{\mathbb{U}_2} \gamma(s,u) \, \widetilde{N}(\mathrm{d}s,\mathrm{d}u) \bigg)^{\top}\tag{35}\] for \(t \in [0,T]\), which, by Assumption 7, defines an \(\mathcal{F}^Y_t\)-adapted local martingale \(G = (G_t)_{t \in [0,T]}\). By Lemma 12, we also have that \(G\) is independent of \(B\) and \(N_1\).
In the following, we will continue to consider \(N_2, \widetilde{W}\) and \(G\) as being defined on \((\Omega,\mathcal{F},\widetilde{\mathbb{P}})\). However, we now introduce a second probability space, which we denote by \((\bar{\Omega},\bar{\mathcal{F}},\bar{\mathbb{P}})\). On this new space, we let \(B\) be a Brownian motion, and let \(N_1\) be a Poisson random measure with compensator \(\nu_1(\mathrm{d}u) \,\mathrm{d}t\). We also define the product space \[(\hat{\Omega},\hat{\mathcal{F}},\hat{\mathbb{P}}) := (\Omega \times \bar{\Omega},\mathcal{F}\otimes \bar{\mathcal{F}},\widetilde{\mathbb{P}}\otimes \bar{\mathbb{P}}).\]
Of course, we can consider all stochastic objects as also living on the product space, by simply letting, e.g., \(B(\omega,\bar{\omega}) = B(\bar{\omega})\) for all \((\omega,\bar{\omega}) \in \hat{\Omega}\). Moreover, by the independence provided in Lemma 12, it is clear that, after this change of framework, the law of these stochastic objects (under the measure \(\hat{\mathbb{P}}\)) is unchanged from how they were defined originally (under \(\widetilde{\mathbb{P}}\)).
Let \(X\) and \(Y\) be the solutions to the SDEs in 34 , now defined on \((\hat{\Omega},\hat{\mathcal{F}},\hat{\mathbb{P}})\). We also define \[\label{eq:32defn32I32on32product32space} \begin{align} I_t &:= \int_0^t h(s,X_s,Y_s)^\top \,\mathrm{d}\widetilde{W}_s - \frac{1}{2} \int_0^t |h(s,X_s,Y_s)|^2 \,\mathrm{d}s + \int_0^t \int_{\mathbb{U}_2} \kappa(s,X_{s-}) \gamma(s,u) \, \widetilde{N}(\mathrm{d}s,\mathrm{d}u)\\ &\quad + \int_0^t \big(1 - \lambda(s,X_{s-},u) + \log(\lambda(s,X_{s-},u))\big) \, \nu_2(\mathrm{d}u) \,\mathrm{d}s\\ &= \int_0^t H(s,X_{s-},Y_{s-}) \,\mathrm{d}G_s - \frac{1}{2} \int_0^t |h(s,X_s,Y_s)|^2 \,\mathrm{d}s\\ &\quad + \int_0^t \big(1 - \lambda(s,X_{s-},u) + \log(\lambda(s,X_{s-},u))\big) \, \nu_2(\mathrm{d}u) \,\mathrm{d}s, \end{align}\tag{36}\] where \[H(t,x,y) := \big(h(t,x,y)^\top, 0, 0, \kappa(t,x)\big).\] Again, it is clear that \((X,Y,I)\) has the same law under \(\hat{\mathbb{P}}\) as it originally had under \(\widetilde{\mathbb{P}}\).
For any (deterministic) càdlàg rough path \(\boldsymbol{\eta} \in \mathscr{V}^p\), we denote by \((X^{\boldsymbol{\eta}}, Y^{\boldsymbol{\eta}}) \in V^p L^{2,\infty}(\bar{\Omega})\) the solution to the rough SDE \[\label{eq:32filtering32RSDE} \begin{align} \mathrm{d}X^{\boldsymbol{\eta}}_t &= \tilde{b}_1(t,X^{\boldsymbol{\eta}}_t,Y^{\boldsymbol{\eta}}_t) \,\mathrm{d}t + \sigma_0(t,X^{\boldsymbol{\eta}}_t,Y^{\boldsymbol{\eta}}_t) \,\mathrm{d}B_t + \int_{\mathbb{U}_1} f_1(t,X^{\boldsymbol{\eta}}_{t-},Y^{\boldsymbol{\eta}}_{t-},u) \, \widetilde{N}_1(\mathrm{d}t,\mathrm{d}u)\\ &\quad + (\sigma_1(t,X^{\boldsymbol{\eta}}_t,Y^{\boldsymbol{\eta}}_t), h_2(t,X^{\boldsymbol{\eta}}_t,Y^{\boldsymbol{\eta}}_t), 0, 0) \,\mathrm{d}\boldsymbol{\eta}_t,\\ \mathrm{d}Y^{\boldsymbol{\eta}}_t &= (\sigma_2(t,Y^{\boldsymbol{\eta}}_t), 0, h_3(t,Y^{\boldsymbol{\eta}}_t),0) \,\mathrm{d}\boldsymbol{\eta}_t, \end{align}\tag{37}\] defined on \((\bar{\Omega},\bar{\mathcal{F}},\bar{\mathbb{P}})\), and we let \(I^{\boldsymbol{\eta}} = I^{1,\boldsymbol{\eta}} + I^{2,\boldsymbol{\eta}}\), where \[\label{eq:32definitio32I94eta} \begin{align} I^{1,\boldsymbol{\eta}}_t &:= \int_0^t H(s,X^{\boldsymbol{\eta}}_s,Y^{\boldsymbol{\eta}}_s) \,\mathrm{d}\boldsymbol{\eta}_s - \frac{1}{2} \int_0^t |h(s,X^{\boldsymbol{\eta}}_s,Y^{\boldsymbol{\eta}}_s)|^2 \,\mathrm{d}s,\\ I^{2,\boldsymbol{\eta}}_t &:= \int_0^t \int_{\mathbb{U}_2} \big(1 - \lambda(s,X^{\boldsymbol{\eta}}_{s-},u) + \log(\lambda(s,X^{\boldsymbol{\eta}}_{s-},u))\big) \, \nu_2(\mathrm{d}u) \,\mathrm{d}s \end{align}\tag{38}\] for \(t \in [0,T]\), noting that these equations are well-defined by Assumptions 8 and 6, combined with the fact that, since \((v - 1)/v \leq \log(v) \leq v - 1\) for all \(v > 0\), we have \[\label{eq:32bound32on321-lambda4332log40lambda41} \big| 1 - \lambda(s,x,u) + \log(\lambda(s,x,u)) \big| = \lambda(s,x,u) - 1 - \log(\lambda(s,x,u)) \leq \frac{(1 - \lambda(s,x,u))^2}{\lambda(s,x,u)}.\tag{39}\]
We now proceed to consider continuity of the conditional distribution, viewed as a function of the rough path driving the system 37 . To this end, given a bounded measurable function \(F \colon \mathbb{R}^{d_X + d_Y} \to \mathbb{R}\), we define functions \(g^F\) and \(\Theta^F\) on the space of rough paths \(\mathscr{V}^p\), such that \[\label{eq:32defn32g94F32Theta94F} g^F_t(\boldsymbol{\eta}) := \bar{\mathbb{E}} \big[ F(X^{\boldsymbol{\eta}}_t,Y^{\boldsymbol{\eta}}_t) \exp (I^{\boldsymbol{\eta}}_t) \big] \qquad \text{and} \qquad \Theta^F_t(\boldsymbol{\eta}) := \frac{g^F_t(\boldsymbol{\eta})}{g^1_t(\boldsymbol{\eta})}\tag{40}\] for each \(\boldsymbol{\eta} \in \mathscr{V}^p\) and \(t \in [0,T]\).
We are now ready to present the first main result of this section.
Theorem 6. Suppose that \(f_1\) satisfies Assumption 1 for \(q = 2\) and some \(p \in [2,3)\), and that Assumptions 6 and 8 also hold. For any \(F \in C^1_b\) and \(\boldsymbol{\eta} \in \mathscr{V}^p\), we have that \(g^F(\boldsymbol{\eta}), \Theta^F(\boldsymbol{\eta}) \in D([0,T];\mathbb{R})\) (the space of real-valued càdlàg paths). Moreover, the following hold.
If \(F \in C^1_b\), and if \(\mathscr{V}^p\) is endowed with (rough path) \(p\)-variation topology, and \(D([0,T];\mathbb{R})\) with the uniform topology, then \(g^F\) and \(\Theta^F\) are both locally Lipschitz continuous.
If \(F \in C^2_b\), then \(g^F(\boldsymbol{\eta}), \Theta^F(\boldsymbol{\eta}) \in V^p([0,T];\mathbb{R})\), and if \(\mathscr{V}^p\) and \(V^p([0,T];\mathbb{R})\) are both endowed with \(p\)-variation topology, then \(g^F\) and \(\Theta^F\) are both locally Lipschitz continuous.
For any \(F \in C^1_b\) and any fixed \(t \in (0,T]\), \(g^F_t\) and \(\Theta^F_t\) are continuous when \(\mathscr{V}^p\) is endowed with the \(p\)-variation J1-Skorokhod distance \(\sigma_{p,[0,t]}\), as defined in 8 .
Proof. We first note that \(I^{\boldsymbol{\eta}} \in \mathrm{BMO}^{p\mathrm{-var}}\) by Proposition 16, and hence that \(\exp(I^{\boldsymbol{\eta}})\) has finite moments of all orders and is also càdlàg in \(L^1\) by Proposition 15. Since \(F\) is Lipschitz, it is also clear that \(F(X^{\boldsymbol{\eta}}, Y^{\boldsymbol{\eta}})\) is càdlàg in \(L^2\). Since, for any \((s,t) \in \Delta_{[0,T]}\), \[\begin{align} |g^F_t(\boldsymbol{\eta}) - g^F_s(\boldsymbol{\eta})| &\leq \bar{\mathbb{E}} \big[ \big|F(X^{\boldsymbol{\eta}}_t, Y^{\boldsymbol{\eta}}_t) \exp(I^{\boldsymbol{\eta}}_t) - F(X^{\boldsymbol{\eta}}_s, Y^{\boldsymbol{\eta}}_s) \exp(I^{\boldsymbol{\eta}}_s)\big| \big]\\ &\leq \bar{\mathbb{E}} \big[ \big| \delta F(X^{\boldsymbol{\eta}}, Y^{\boldsymbol{\eta}})_{s,t} \exp(I^{\boldsymbol{\eta}}_t) \big| \big] + \bar{\mathbb{E}} \big[ \big| F(X^{\boldsymbol{\eta}}_s, Y^{\boldsymbol{\eta}}_s) \delta \exp(I^{\boldsymbol{\eta}})_{s,t} \big| \big]\\ &\lesssim \|\delta F(X^{\boldsymbol{\eta}}, Y^{\boldsymbol{\eta}})_{s,t}\|_{L^2(\bar{\Omega})} + \|\delta \exp(I^{\boldsymbol{\eta}})_{s,t}\|_{L^1(\bar{\Omega})}, \end{align}\] it follows that \(g^F(\boldsymbol{\eta})\), and hence also \(\Theta^F(\boldsymbol{\eta})\), are right-continuous. Similarly, we deduce that, whenever \(s_n \nearrow s\), the sequence \((g^F_{s_n}(\boldsymbol{\eta}))_{n \in \mathbb{N}}\) is Cauchy, so that we also have the existence of left-limits.
(i): Let \(\boldsymbol{\eta}, \tilde{\boldsymbol{\eta}} \in \mathscr{V}^p\) with \(\|\boldsymbol{\eta}\|_{p,[0,T]}, \| \tilde{\boldsymbol{\eta}}\|_{p,[0,T]} \leq L\) for some constant \(L > 0\). Then, by Hölder’s inequality and the inequality \(|\exp(x) - \exp(y)| \leq |x-y| (\exp(x) \vee \exp(y))\), we have, for any \(t \in [0,T]\), \[\label{eq:32first32Lipschitz32bound32on32g94F} \begin{align} |g^F_t(\boldsymbol{\eta}) - g^F_t(\tilde{\boldsymbol{\eta}})| &\leq \bar{\mathbb{E}} \big[ \big| F(X^{\boldsymbol{\eta}}_t, Y^{\boldsymbol{\eta}}_t) \exp(I^{\boldsymbol{\eta}}_t) - F(X^{\tilde{\boldsymbol{\eta}}}_t, Y^{\tilde{\boldsymbol{\eta}}}_t) \exp(I^{\tilde{\boldsymbol{\eta}}}_t) \big| \big]\\ &\leq \big\| F(X^{\boldsymbol{\eta}}_t, Y^{\boldsymbol{\eta}}_t) - F(X^{\tilde{\boldsymbol{\eta}}}_t, Y^{\tilde{\boldsymbol{\eta}}}_t) \big\|_{L^2(\bar{\Omega})} \| \exp(I^{\tilde{\boldsymbol{\eta}}}_t) \|_{L^2(\bar{\Omega})}\\ &\quad + \|F\|_\infty \big\| \exp(I^{\boldsymbol{\eta}}_t) \vee \exp(I^{\tilde{\boldsymbol{\eta}}}_t) \big\|_{L^2(\bar{\Omega})} \| I^{\boldsymbol{\eta}}_t - I^{\tilde{\boldsymbol{\eta}}}_t \|_{L^2(\bar{\Omega})}. \end{align}\tag{41}\] Using the Lipschitz continuity of \(F\), and the stability of solutions to rough SDEs 7 , we have that \[\big\| F(X^{\boldsymbol{\eta}}_t, Y^{\boldsymbol{\eta}}_t) - F(X^{\tilde{\boldsymbol{\eta}}}_t, Y^{\tilde{\boldsymbol{\eta}}}_t) \big\|_{L^2(\bar{\Omega})} \lesssim \| X^{\boldsymbol{\eta}}_t - X^{\tilde{\boldsymbol{\eta}}}_t \|_{L^2(\bar{\Omega})} + \| Y^{\boldsymbol{\eta}}_t - Y^{\tilde{\boldsymbol{\eta}}}_t \|_{L^2(\bar{\Omega})} \lesssim \|\boldsymbol{\eta} - \tilde{\boldsymbol{\eta}}\|_{p,[0,T]}.\] The difference \(\|I^{1,\boldsymbol{\eta}}_t - I^{1,\tilde{\boldsymbol{\eta}}}_t\|_{L^2(\bar{\Omega})}\) may be similarly bounded using the stability of rough stochastic integration ([22]) and of solutions to rough SDEs again. Finally, by Assumption 8, we have that \[\label{eq:32Lipschitz32continuity32of32lambda} \begin{align} |&\lambda(s,X^{\boldsymbol{\eta}}_{s-},u) - \lambda(s,X^{\tilde{\boldsymbol{\eta}}}_{s-},u)|\\ &\leq |\lambda(s,X^{\boldsymbol{\eta}}_{s-},u) \vee \lambda(s,X^{\tilde{\boldsymbol{\eta}}}_{s-},u)| |\log(\lambda(s,X^{\boldsymbol{\eta}}_{s-},u)) - \log(\lambda(s,X^{\tilde{\boldsymbol{\eta}}}_{s-},u))|\\ &= |\lambda(s,X^{\boldsymbol{\eta}}_{s-},u) \vee \lambda(s,X^{\tilde{\boldsymbol{\eta}}}_{s-},u)| |\kappa(s,X^{\boldsymbol{\eta}}_{s-}) - \kappa(s,X^{\tilde{\boldsymbol{\eta}}}_{s-})| |\gamma(s,u)|\\ &\lesssim \|\kappa\|_{C^1_b} |X^{\boldsymbol{\eta}}_{s-} - X^{\tilde{\boldsymbol{\eta}}}_{s-}| |\gamma(s,u)|, \end{align}\tag{42}\] so that \[\label{eq:32I94232eta32bound} \|I^{2,\boldsymbol{\eta}}_t - I^{2,\tilde{\boldsymbol{\eta}}}_t\|_{L^2(\bar{\Omega})} \lesssim \bigg(\int_0^T \int_{\mathbb{U}_2} |\gamma(s,u)| \, \nu_2(\mathrm{d}u) \,\mathrm{d}s\bigg) \|X^{\boldsymbol{\eta}} - X^{\tilde{\boldsymbol{\eta}}}\|_{p,2,[0,T],\bar{\Omega}} \lesssim \|X^{\boldsymbol{\eta}} - X^{\tilde{\boldsymbol{\eta}}}\|_{p,2,[0,T],\bar{\Omega}},\tag{43}\] and it follows that \(\sup_{t \in [0,T]} |g^F_t(\boldsymbol{\eta}) - g^F_t(\tilde{\boldsymbol{\eta}})| \lesssim \|\boldsymbol{\eta} - \tilde{\boldsymbol{\eta}}\|_{p,[0,T]}\). By Jensen’s inequality, we have that \[\label{eq:32bound32on32g94140eta41} g^1_t(\boldsymbol{\eta}) = \bar{\mathbb{E}} [\exp(I^{\boldsymbol{\eta}}_t)] \geq \exp(-\bar{\mathbb{E}} [|I^{\boldsymbol{\eta}}_t|]),\tag{44}\] from which we infer by Proposition 16 that \(g^1_t(\boldsymbol{\eta})\) is bounded from both above and below (with bounds which depend on \(L\)). It follows that \(g^F\) and \(\Theta^F\) are both locally Lipschitz.
(ii): Let \(\boldsymbol{\eta}, \tilde{\boldsymbol{\eta}} \in \mathscr{V}^p\) such that \(\|\boldsymbol{\eta}\|_{p,[0,T]}, \|\tilde{\boldsymbol{\eta}}\|_{p,[0,T]} \leq L\) for some constant \(L > 0\). For \((s,t) \in \Delta_{[0,T]}\), we have that \[\begin{align} |\delta g^F_{s,t}(\boldsymbol{\eta}) - \delta g^F_{s,t}(\tilde{\boldsymbol{\eta}})| &\leq \bar{\mathbb{E}} \big[ \big| \delta F(X^{ \boldsymbol{\eta}},Y^{ \boldsymbol{\eta}})_{s,t} \exp(I^{\boldsymbol{\eta}}_t) - \delta F(X^{\tilde{\boldsymbol{\eta}}},Y^{\tilde{\boldsymbol{\eta}}})_{s,t} \exp(I^{\tilde{\boldsymbol{\eta}}}_t) \big| \big]\\ &\quad + \bar{\mathbb{E}} \big[ \big| F(X^{\boldsymbol{\eta}},Y^{\boldsymbol{\eta}})_s \delta \exp(I^{\boldsymbol{\eta}})_{s,t} - F(X^{\tilde{\boldsymbol{\eta}}},Y^{\tilde{\boldsymbol{\eta}}})_s \delta \exp(I^{\tilde{\boldsymbol{\eta}}})_{s,t} \big| \big]\\ &\leq \bar{\mathbb{E}} \big[ \big| \delta F(X^{\boldsymbol{\eta}},Y^{\boldsymbol{\eta}})_{s,t} - \delta F(X^{\tilde{\boldsymbol{\eta}}},Y^{\tilde{\boldsymbol{\eta}}})_{s,t} \big| \big| \exp(I^{\boldsymbol{\eta}}_t) \big| \big]\\ &\quad + \bar{\mathbb{E}} \big[ \big| \delta F(X^{\tilde{\boldsymbol{\eta}}},Y^{\tilde{\boldsymbol{\eta}}})_{s,t} \big| \big| \delta \exp(I^{\boldsymbol{\eta}})_{s,t} - \delta \exp(I^{\tilde{\boldsymbol{\eta}}})_{s,t} \big| \big]\\ &\quad + \bar{\mathbb{E}} \big[ \big| \delta F(X^{\tilde{\boldsymbol{\eta}}},Y^{\tilde{\boldsymbol{\eta}}})_{s,t} \big| \big| \exp(I^{\boldsymbol{\eta}}_s) - \exp(I^{\tilde{\boldsymbol{\eta}}}_s) \big| \big]\\ &\quad + \bar{\mathbb{E}} \big[ \big| F(X^{\boldsymbol{\eta}},Y^{\boldsymbol{\eta}})_s - F(X^{\tilde{\boldsymbol{\eta}}},Y^{\tilde{\boldsymbol{\eta}}})_s \big| \big| \delta \exp(I^{\boldsymbol{\eta}})_{s,t} \big| \big]\\ &\quad + \bar{\mathbb{E}} \big[ \big| F(X^{\tilde{\boldsymbol{\eta}}},Y^{\tilde{\boldsymbol{\eta}}})_s \big| \big| \delta \exp(I^{\boldsymbol{\eta}})_{s,t} - \delta \exp(I^{\tilde{\boldsymbol{\eta}}})_{s,t} \big| \big]. \end{align}\] By Hölder’s inequality, Proposition 18, and an application of Proposition 15 to \(I^{\boldsymbol{\eta}}, I^{\tilde{\boldsymbol{\eta}}}\) (with \(\tilde{q} = r = 1\) and \(q = 2\)), we then have that \[\begin{align} |\delta g^F_{s,t}(\boldsymbol{\eta}) - \delta g^F_{s,t}(\tilde{\boldsymbol{\eta}})| &\lesssim \|X^{\boldsymbol{\eta}} - X^{\tilde{\boldsymbol{\eta}}}\|_{p,2,[s,t],\bar{\Omega}} + \|Y^{\boldsymbol{\eta}} - Y^{\tilde{\boldsymbol{\eta}}}\|_{p,2,[s,t],\bar{\Omega}}\\ &\quad + \|F\|_\infty \|I^{\boldsymbol{\eta}} - I^{\tilde{\boldsymbol{\eta}}}\|_{p,2,[s,t],\bar{\Omega}}\\ &\quad +\big(\|X^{\boldsymbol{\eta}}\|_{p,2,\infty,[s,t],\bar{\Omega}} + \|Y^{\boldsymbol{\eta}}\|_{p,2,\infty,[s,t],\bar{\Omega}}\big) \|I^{\boldsymbol{\eta}} - I^{\tilde{\boldsymbol{\eta}}}\|_{p,2,[0,T],\bar{\Omega}}\\ &\quad + \big( \|X^{\boldsymbol{\eta}} - X^{\tilde{\boldsymbol{\eta}}}\|_{p,2,[0,T],\bar{\Omega}} + \|Y^{\boldsymbol{\eta}} - Y^{\tilde{\boldsymbol{\eta}}}\|_{p,2,[0,T],\bar{\Omega}} \big) \|\exp(I^{\boldsymbol{\eta}})\|_{p,2,[s,t],\bar{\Omega}}\\ &\quad + \|F\|_\infty \|I^{\boldsymbol{\eta}} - I^{\tilde{\boldsymbol{\eta}}}\|_{p,2,[s,t],\bar{\Omega}}, \end{align}\] where we also used the first bound in [22] (which requires \(F \in C^2_b\)) to bound \(\|F(X^{\boldsymbol{\eta}}, Y^{\boldsymbol{\eta}}) - F(X^{\tilde{\boldsymbol{\eta}}}, Y^{\tilde{\boldsymbol{\eta}}})\|_{p,2,[s,t],\bar{\Omega}} \lesssim \|X^{\boldsymbol{\eta}} - X^{\tilde{\boldsymbol{\eta}}}\|_{p,2,[s,t],\bar{\Omega}} + \|Y^{\boldsymbol{\eta}} - Y^{\tilde{\boldsymbol{\eta}}}\|_{p,2,[s,t],\bar{\Omega}}\).
We can bound \(\|I^{1,\boldsymbol{\eta}} - I^{1,\tilde{\boldsymbol{\eta}}}\|_{p,2,[0,T],\bar{\Omega}} \lesssim \|\boldsymbol{\eta} - \tilde{\boldsymbol{\eta}}\|_{p,[0,T]}\) using the stability of rough integration ([22]) and of solutions to rough SDEs 7 , and we can use 42 again to see that \[\|I^{2,\boldsymbol{\eta}} - I^{2,\tilde{\boldsymbol{\eta}}}\|_{p,2,[0,T],\bar{\Omega}} \lesssim \bigg( \int_0^T \int_{\mathbb{U}_2} |\gamma(r,u)| \, \nu_2(\mathrm{d}u) \,\mathrm{d}r \bigg) \|X^{\boldsymbol{\eta}} - X^{\tilde{\boldsymbol{\eta}}}\|_{p,2,[0,T],\bar{\Omega}} \lesssim \|\boldsymbol{\eta} - \tilde{\boldsymbol{\eta}}\|_{p,[0,T]}.\] It follows that \(\|g^F(\boldsymbol{\eta}) - g^F(\tilde{\boldsymbol{\eta}})\|_{p,[0,T]} \lesssim \|\boldsymbol{\eta} - \tilde{\boldsymbol{\eta}}\|_{p,[0,T]}\).
A straightforward calculation, using the fact that \(g^1_t(\boldsymbol{\eta})\) is bounded below by 44 , shows that \[\begin{align} |\delta \Theta^F_{s,t}(\boldsymbol{\eta}) - \delta \Theta^F_{s,t}(\tilde{\boldsymbol{\eta}})| &\lesssim |\delta g^F_{s,t}(\boldsymbol{\eta}) - \delta g^F_{s,t}(\tilde{\boldsymbol{\eta}})| + |g^1_t(\boldsymbol{\eta}) - g^1_t(\tilde{\boldsymbol{\eta}})| |\delta g^F_{s,t}(\tilde{\boldsymbol{\eta}})|\\ &\quad + |g^1_t(\tilde{\boldsymbol{\eta}}) g^1_s(\tilde{\boldsymbol{\eta}}) g^F_s(\boldsymbol{\eta}) \delta g^1_{s,t}(\boldsymbol{\eta}) - g^1_t(\boldsymbol{\eta}) g^1_s(\boldsymbol{\eta}) g^F_s(\tilde{\boldsymbol{\eta}}) \delta g^1_{s,t}(\tilde{\boldsymbol{\eta}})|. \end{align}\] We have already shown how to bound the first term on the right-hand side. For the second term, we may use the result of part (i), and the fact that \(\|g^F(\tilde{\boldsymbol{\eta}})\|_{p,[0,T]}\) is uniformly bounded (by a constant which depends in particular on \(F\) and \(L\)), and we can similarly bound the final term by further splitting it up by adding zeros in the obvious way. Putting this all together, we deduce that \(\|\Theta^F(\boldsymbol{\eta}) - \Theta^F(\tilde{\boldsymbol{\eta}})\|_{p,[0,T]} \lesssim \|\boldsymbol{\eta} - \tilde{\boldsymbol{\eta}}\|_{p,[0,T]}\).
(iii): Let \(\boldsymbol{\eta} \in \mathscr{V}^p\) and \(t \in (0,T]\), and let \((\boldsymbol{\eta}^n)_{n \in \mathbb{N}} \subset \mathscr{V}^p\) be a sequence of rough paths such that \(\sigma_{p,[0,t]}(\boldsymbol{\eta}^n,\boldsymbol{\eta}) \to 0\) as \(n \to \infty\). In the proof of part (i) above, we showed in particular that \[|g^F_t(\boldsymbol{\eta}^n) - g^F_t(\boldsymbol{\eta})| \lesssim \| X^{\boldsymbol{\eta}^n}_t - X^{\boldsymbol{\eta}}_t \|_{L^2(\bar{\Omega})} + \| Y^{\boldsymbol{\eta}^n}_t - Y^{\boldsymbol{\eta}}_t \|_{L^2(\bar{\Omega})} + \| I^{\boldsymbol{\eta}^n}_t - I^{\boldsymbol{\eta}}_t \|_{L^2(\bar{\Omega})}.\] It follows from Proposition 2 and Remark 4 that the right-hand side above tends to zero as \(n \to \infty\), and we deduce that \(g^F_t\), and hence also \(\Theta^F_t\), are continuous. ◻
As highlighted by, e.g., Crisan et al. [13], a benefit of obtaining a robust representation of the conditional distribution is that in real-world applications the model chosen for the observation process may be an imperfect one, but a robust representation ensures that, provided the chosen model is close in some weak sense to the real one, our estimate of the conditional distribution should still be close to the true distribution. The following result can be interpreted as a novel and concrete formulation of this idea.
Theorem 7. Let \(m \in [1,\infty)\), \(\varepsilon> 0\), \(\alpha > 0\), \(L > 0\) and \(F \in C^1_b\), and let us adopt the assumptions of Theorem 6. There exist constants \(\beta > 0\) and \(C > 0\), which depend on \(p, T, m, \varepsilon, \alpha\) and \(\|F\|_{C^1_b}\), as well as the constants in Assumption 1 for \(f_1\), and on the other coefficients and constants specified in Assumption 8 (and \(C\) also depends on \(L\)), such that, whenever \(\mathbf{V}\) and \(\widetilde{\mathbf{V}}\) are two random càdlàg rough paths on \((\Omega,\mathcal{F},\widetilde{\mathbb{P}})\) which satisfy \[\label{eq:32exp32needs32to32be32L1} \Big\| \exp \Big( \beta \Big( N_{\alpha,[0,T]}\big(\|\mathbf{V}\|_{p,[\cdot,\cdot)}^p\big) + 1 \Big) \Big( 1 + \sup_{t \in (0,T]} |\Delta \mathbf{V}_t| \Big) \Big) \Big\|_{L^1(\Omega)} \leq L,\tag{45}\] and the same inequality with \(\mathbf{V}\) replaced by \(\widetilde{\mathbf{V}}\), we then have that \[\label{eq:32local32Lipschitz32continuity32in32L94m32for32Theta} \Big\| \sup_{t \in [0,T]} \big|\Theta^F_t(\mathbf{V}) - \Theta^F_t(\widetilde{\mathbf{V}})\big| \Big\|_{L^m(\Omega)} \leq C \big\| \|\mathbf{V}- \widetilde{\mathbf{V}}\|_{p,[0,T]} \big\|_{L^{m+\varepsilon}(\Omega)}.\tag{46}\]
Proof. Let \(n > 1\) such that \(\frac{1}{n} + \frac{1}{m+\varepsilon} = \frac{1}{m+\frac{\varepsilon}{2}}\). Recalling the bound in 41 , we have that \[\begin{align} &\Big\| \sup_{t \in [0,T]} \big|g^F_t(\mathbf{V}) - g^F_t(\widetilde{\mathbf{V}})\big| \Big\|_{L^{m+\frac{\varepsilon}{2}}(\Omega)}\\ &\lesssim \Big\| \sup_{t \in [0,T]} \big\| F(X^{\mathbf{V}}_t,Y^{\mathbf{V}}_t) - F(X^{\widetilde{\mathbf{V}}}_t,Y^{\widetilde{\mathbf{V}}}_t) \big\|_{L^2(\bar{\Omega})} \Big\|_{L^{m+\varepsilon}(\Omega)} \Big\| \sup_{t \in [0,T]} \|\exp(I^{\widetilde{\mathbf{V}}}_t)\|_{L^2(\bar{\Omega})} \Big\|_{L^n(\Omega)}\\ &\quad + \Big\| \sup_{t \in [0,T]} \|\exp(I^{\mathbf{V}}_t) \vee \exp(I^{\widetilde{\mathbf{V}}}_t)\|_{L^2(\bar{\Omega})} \Big\|_{L^n(\Omega)} \Big\| \sup_{t \in [0,T]} \|I^{\mathbf{V}}_t - I^{\widetilde{\mathbf{V}}}_t\|_{L^2(\bar{\Omega})} \Big\|_{L^{m+\varepsilon}(\Omega)}. \end{align}\] Let \(I^{\mathbf{V}} = I^{1,\mathbf{V}} + I^{2,\mathbf{V}}\), as in 38 . By the estimate in ?? and the assumption in 45 , for any \(q \in [1,\infty)\), there exists a constant \(\zeta > 0\) such that \[\Big\| \exp \Big( q \sup_{t \in [0,T]} |I^{1,\mathbf{V}}_t| \Big) \Big\|_{L^1(\bar{\Omega})} \leq \exp \Big( \zeta \Big( N_{\alpha,[0,T]}\big(\|\mathbf{V}\|_{p,[\cdot,\cdot)}^p\big) + 1 \Big) \Big( 1 + \sup_{t \in (0,T]} |\Delta \mathbf{V}_t| \Big) \Big),\] and so by choosing \(\beta = \zeta\) in 45 , we have that \(\| \| \exp ( q \sup_{t \in [0,T]} |I^{1,\mathbf{V}}_t| ) \|_{L^1(\bar{\Omega})} \|_{L^1(\Omega)} \leq L\). Recalling 39 and Assumption 6, we also have that \[\Big\| \exp \Big( q \sup_{t \in [0,T]} |I^{2,\mathbf{V}}_t| \Big) \Big\|_{L^\infty(\bar{\Omega})} \leq \exp \bigg( q T \sup_{t \in [0,T], \, x \in \mathbb{R}^{d_X}} \int_{\mathbb{U}_2} \frac{(1 - \lambda(t,x,u))^2}{\lambda(t,x,u)} \, \nu_2(\mathrm{d}u) \bigg) < \infty,\] and the above also holds for \(I^{\widetilde{\mathbf{V}}} = I^{1,\widetilde{\mathbf{V}}} + I^{2,\widetilde{\mathbf{V}}}\). We then have that \[\begin{align} \Big\| \sup_{t \in [0,T]} \big|g^F_t(\mathbf{V}) - g^F_t(\widetilde{\mathbf{V}})\big| \Big\|_{L^{m+\frac{\varepsilon}{2}}(\Omega)} &\lesssim \Big\| \sup_{t \in [0,T]} \big\|F(X^{\mathbf{V}}_t,Y^{\mathbf{V}}_t) - F(X^{\widetilde{\mathbf{V}}}_t,Y^{\widetilde{\mathbf{V}}}_t)\big\|_{L^2(\bar{\Omega})} \Big\|_{L^{m+\varepsilon}(\Omega)}\\ &\quad + \Big\| \sup_{t \in [0,T]} \|I^{\mathbf{V}}_t - I^{\widetilde{\mathbf{V}}}_t\|_{L^2(\bar{\Omega})} \Big\|_{L^{m+\varepsilon}(\Omega)}. \end{align}\] The first term above can be bounded by the right-hand side of 46 by the Lipschitz continuity of \(F\), and an application of Proposition 9, which is applicable since the combination of Lemma 17 with the assumption in 45 ensures that \(\|\mathbf{V}\|_{p,[0,T]}\) and \(\|\widetilde{\mathbf{V}}\|_{p,[0,T]}\) have finite moments of all orders. For the second term, we again split \(I^{\mathbf{V}} - I^{\widetilde{\mathbf{V}}}\) into the sum of \(I^{1,\mathbf{V}} - I^{1,\widetilde{\mathbf{V}}}\) and \(I^{2,\mathbf{V}} - I^{2,\widetilde{\mathbf{V}}}\), where the former may be treated using [22] and Proposition 9, while the latter may be treated by recalling 43 and again using Proposition 9. We thus obtain \[\Big\| \sup_{t \in [0,T]} \big|g^F_t(\mathbf{V}) - g^F_t(\widetilde{\mathbf{V}})\big| \Big\|_{L^{m+\frac{\varepsilon}{2}}(\Omega)} \lesssim \big\| \|\mathbf{V}- \widetilde{\mathbf{V}}\|_{p,[0,T]} \big\|_{L^{m+\varepsilon}(\Omega)}.\]
Let \(\tilde{n} > 1\) such that \(\frac{1}{\tilde{n}} + \frac{1}{m+\frac{\varepsilon}{2}} = \frac{1}{m}\). By Hölder’s inequality, it is straightforward to see that \[\label{eq:32Theta32V32-32Theta32tV32bound} \begin{align} \bigg\| &\sup_{t \in [0,T]} \big|\Theta^F_t(\mathbf{V}) - \Theta^F_t(\widetilde{\mathbf{V}})\big| \bigg\|_{L^m(\Omega)}\\ &\leq \bigg\| \sup_{t \in [0,T]} \big|g^F_t(\mathbf{V}) - g^F_t(\widetilde{\mathbf{V}})\big| \bigg\|_{L^{m+\frac{\varepsilon}{2}}(\Omega)} \bigg\| \sup_{t \in [0,T]} \frac{1}{g^1_t(\mathbf{V})} \bigg\|_{L^{\tilde{n}}(\Omega)}\\ &\quad + \bigg\| \sup_{t \in [0,T]} \big|g^1_t(\mathbf{V}) - g^1_t(\widetilde{\mathbf{V}})\big| \bigg\|_{L^{m+\frac{\varepsilon}{2}}(\Omega)} \bigg\| \sup_{t \in [0,T]} \frac{|g^F_t(\widetilde{\mathbf{V}})|}{g^1_t(\mathbf{V}) g^1_t(\widetilde{\mathbf{V}})} \bigg\|_{L^{\tilde{n}}(\Omega)}. \end{align}\tag{47}\] By Jensen’s inequality, we have that \[\sup_{t \in [0,T]} \frac{1}{g^1_t(\mathbf{V})} \leq \frac{1}{\bar{\mathbb{E}}[\exp(-\sup_{t \in [0,T]} |I^{\mathbf{V}}_t|)]} \leq \frac{1}{\exp(\bar{\mathbb{E}}[-\sup_{t \in [0,T]} |I^{\mathbf{V}}_t|])} \leq \bar{\mathbb{E}} \Big[ \exp \Big( \sup_{t \in [0,T]} |I^{\mathbf{V}}_t| \Big) \Big],\] and since \(F\) is bounded we also have that \(g^F_t(\mathbf{V}) \leq \bar{\mathbb{E}}[\exp(\sup_{t \in [0,T]} |I^{\mathbf{V}}_t|)]\), and of course these inequalities also hold for \(\widetilde{\mathbf{V}}\). By the bounds on \(I^{1,\mathbf{V}}\) and \(I^{2,\mathbf{V}}\) established above, it follows that the norms on the right-hand side of 47 are finite, and we thus obtain the estimate in 46 . ◻
Example 2. If \(\mathbf{V}\) is continuous and satisfies \[\big\| \exp \big( \gamma N_{\alpha,[0,T]}\big(\|\mathbf{V}\|_{p,[\cdot,\cdot)}^p\big)^q \big) \big\|_{L^1(\Omega)} < \infty\] for some \(\alpha > 0\), \(\gamma > 0\) and \(q > 1\), then, for any \(\beta > 0\), one can find an \(L > 0\) for which 45 holds. To see this we simply note that \[\begin{align} \exp \big( \beta N_{\alpha,[0,T]}\big(\|\mathbf{V}\|_{p,[\cdot,\cdot)}^p\big) \big) &\lesssim \exp \big( \beta N_{\alpha,[0,T]}\big(\|\mathbf{V}\|_{p,[\cdot,\cdot)}^p\big) \mathbf{1}_{\{N_{\alpha,[0,T]}(\|\mathbf{V}\|_{p,[\cdot,\cdot)}^p)^{q-1} > \frac{\beta}{\gamma}\}} \big)\\ &\leq \exp \big( \gamma N_{\alpha,[0,T]}\big(\|\mathbf{V}\|_{p,[\cdot,\cdot)}^p\big)^q \big), \end{align}\] where the implicit multiplicative constant depends only on \(\gamma, q\) and \(\beta\).
By [17], we thus conclude that, if \(V\) is a centered Gaussian process with covariance of finite mixed \((1,\rho)\)-variation for some \(\rho < 2\), then the enhanced Gaussian process \(\mathbf{V}\) (as defined in [16]) satisfies 45 . In particular, this includes the case of fractional Brownian motion with Hurst parameter \(H > \frac{1}{4}\); see [55]. In [54], the authors provide another example of a random rough path fulfilling 45 , namely certain Markovian rough paths emerging from specific Dirichlet forms.
Remark 21. Suppose that \((\mathbf{V}^n)_{n \in \mathbb{N}}\) is a sequence of approximations of the random rough path \(\mathbf{V}\), such that convergence rates for \(\mathbf{V}^n \to \mathbf{V}\) are known in a suitable Lebesgue space \(L^{m+\varepsilon}\). By Theorem 7, those convergence rates then carry over to the \(L^m\) approximation of \(\Theta^F_t(\mathbf{V})\) by \(\Theta^F_t(\mathbf{V}^n)\). For example, if \(\mathbf{V}\) is (Stratonovich) Brownian rough path, and \(\mathbf{V}^n\) is the piecewise linear approximation thereof on the \(n\)th dyadic partition, then it follows by [16] that, for any \(\eta \in (0,\frac{1}{2})\), we have the convergence rate \[\Big\| \sup_{t \in [0,T]} \big|\Theta^F_t(\mathbf{V}^n) - \Theta^F_t(\mathbf{V})\big| \Big\|_{L^m(\Omega)} \lesssim \big\| \|\mathbf{V}^n - \mathbf{V}\|_{p,[0,T]} \big\|_{L^{m+\varepsilon}(\Omega)} \lesssim 2^{-\frac{\eta n}{2}}.\]
We recall the local martingale \(G\) defined in 35 . Let us now denote by \(\mathbf{G}= (G,\mathbb{G})\) the Itô rough path lift of \(G\), so that \(\mathbb{G}_{s,t} = \int_s^t \delta G_{s,u} \otimes \mathrm{d}G_u\), defined as an Itô integral on \((\Omega,\mathcal{F},\widetilde{\mathbb{P}})\) for each \((s,t) \in \Delta_{[0,T]}\). Of course, \(\mathbf{G}\) is then a random càdlàg rough path, such that \(\mathbf{G}(\omega) \in \mathscr{V}^p\) for \(\widetilde{\mathbb{P}}\)-almost every \(\omega \in \Omega\) and any \(p \in (2,3)\).
As before, we let \((X,Y)\) be the solution to the (doubly) SDE 34 , and let \(I\) be as defined in 36 , both defined on the product probability space \((\hat{\Omega},\hat{\mathcal{F}},\hat{\mathbb{P}})\). By Theorem 3 (and Remark 8), we have that \((X,Y)\) is also the solution to the corresponding random rough SDE. That is, for \(\widetilde{\mathbb{P}}\)-almost every \(\omega \in \Omega\), we have that \((X(\omega,\cdot), Y(\omega,\cdot))\) is the solution to the rough SDE in 37 driven by \(\boldsymbol{\eta} = \mathbf{G}(\omega)\), i.e., \[\label{eq:32consistency32of32X32and32Y} (X(\omega,\cdot), Y(\omega,\cdot)) = (X^{\mathbf{G}(\omega)}, Y^{\mathbf{G}(\omega)}).\tag{48}\] By Proposition 7, we then also have that \(I\) is the randomised version of the process \(I^{\boldsymbol{\eta}}\) in 38 . That is, for \(\widetilde{\mathbb{P}}\)-almost every \(\omega \in \Omega\), we have that \(I(\omega,\cdot) = I^{\mathbf{G}(\omega)}\).
We recall from Section 5.2 (with a slight abuse of notation) that \((X,Y,I)\) has the same law on \((\Omega,\mathcal{F})\) under \(\widetilde{\mathbb{P}}\) as it did on \((\hat{\Omega},\hat{\mathcal{F}})\) under \(\hat{\mathbb{P}}\). Hence, for any \(t \in [0,T]\) and \(A \in \mathcal{F}^Y_t\), using Fubini’s theorem, and the fact that \(\mathcal{F}^Y_t\) and \(\mathcal{F}^B_t \vee \mathcal{F}^{\widetilde{N}_1}_t\) are independent by Assumption 7, we have that \[\begin{align} \widetilde{\mathbb{E}} \big[F(X_t,Y_t) \exp(I_t) \mathbf{1}_A\big] = \hat{\mathbb{E}} \big[F(X_t,Y_t) \exp(I_t) \mathbf{1}_A\big] = \widetilde{\mathbb{E}} \big[\bar{\mathbb{E}} \big[F(X_t,Y_t) \exp(I_t)\big] \mathbf{1}_A\big]. \end{align}\] By another use of Fubini’s theorem, we also have that \(\bar{\mathbb{E}}[F(X_t,Y_t) \exp(I_t)]\) is \(\mathcal{F}^Y_t\)-measurable, and thus coincides with \(\widetilde{\mathbb{E}}[F(X_t,Y_t) \exp(I_t) \,|\, \mathcal{F}^Y_t]\). By the Kallianpur–Striebel formula (Lemma 13), we then have that the conditional distribution on the product space can be represented as \[\pi_t(F) = \frac{\rho_t(F)}{\rho_t(1)} = \frac{\bar{\mathbb{E}}[F(X_t,Y_t) \exp(I_t)]}{\bar{\mathbb{E}}[\exp(I_t)]}\] for all bounded measurable functions \(F\).
Proposition 22. Suppose that \(f_1\) satisfies Assumption 1 for \(q = 2\) and some \(p \in (2,3)\), and that Assumptions 2–6, 7 and 8 also hold. Let us recall the functions \(g^F\) and \(\Theta^F\) defined in 40 . For any bounded measurable function \(F\), we have that \(g^F(\mathbf{G})\) and \(\Theta^F(\mathbf{G})\) are \(\mathcal{F}^Y_t\)-adapted, and that, for any \(t \in [0,T]\), \[\label{eq:32consistency32result} g^F_t(\mathbf{G}) = \rho_t(F) \quad \text{and} \quad \Theta^F_t(\mathbf{G}) = \pi_t(F)\qquad{(5)}\] hold \(\widetilde{\mathbb{P}}\)-almost surely.
Proof. For \(\widetilde{\mathbb{P}}\)-almost any \(\omega \in \Omega\), using 48 , we have that \[\begin{align} g^F_t(\mathbf{G}(\omega)) &= \bar{\mathbb{E}} \big[ F\big(X^{\mathbf{G}(\omega)}_t,Y^{\mathbf{G}(\omega)}_t\big) \exp \big(I^{\mathbf{G}(\omega)}_t\big) \big]\\ &= \bar{\mathbb{E}} \big[ F(X_t(\omega,\cdot), Y_t(\omega,\cdot)) \exp (I_t(\omega,\cdot)) \big] = \rho_t(F)(\omega). \end{align}\] It follows that the equalities in ?? hold. In particular, we infer that \(g^F_t(\mathbf{G})\) and \(\Theta^F_t(\mathbf{G})\) are \(\mathcal{F}^Y_t\)-measurable random variables. ◻
Remark 23. For any \(F \in C^1_b\), it follows from Theorem 6 and Proposition 22 that \(\Theta^F(\mathbf{G})\) is a càdlàg \(\mathcal{F}^Y_t\)-adapted (and hence \(\mathcal{F}^Y_t\)-optional) process. Thus, although we only defined \(\pi_t(F)\) for fixed times \(t \in [0,T]\), we immediately deduce that \(\pi(F)\) has an \(\mathcal{F}^Y_t\)-optional version (without needing to take an optional projection, as in classical filtering theory).
To express the conditional distribution of the signal as a function of the observable quantities, it was necessary to isolate these terms via a change of measure. As a result, the corresponding Radon–Nikodym derivative involves stochastic integrals against both the continuous and jump components of the observation noise, each with a distinct integrand.
In the continuous additive noise setting of [13], the role of the process \(G\) (as defined in 35 ) is played by the observation process \(Y\), and the robust representation is written as a function of its rough path lift. However, as we have seen, contrary to the continuous setting, obtaining a robust representation in the presence of additional jump terms requires us to observe all the components of \(G\), and to express the filter as a function of the joint lift of these components. In particular, in general it is necessary to be able to distinguish the continuous and jump components of the observation process. This requirement corresponds to Assumption 7, which is commonly adopted in the study of stochastic filtering for jump-diffusion models (see, e.g., [36], [40] or [31]).
While in general one needs to consider the separate jump components of the process \(G\), in the special case when the observation is perturbed by additive noise, it is sufficient to observe only the continuous and jump parts of \(Y\). This is made precise in the following corollary of Theorem 6 and Proposition 22.
Corollary 2. Suppose that \(X\) and \(Y\) satisfy \[\begin{align} \mathrm{d}X_t &= b_1(t,X_t,Y_t) \,\mathrm{d}t + \sigma_0(t,X_t,Y_t) \,\mathrm{d}B_t + \sigma_1(t,X_t,Y_t) \,\mathrm{d}W_t\\ &\quad + \int_{\mathbb{U}_1} f_1(t,X_{t-},Y_{t-},u) \, \widetilde{N}_1(\mathrm{d}t,\mathrm{d}u) + \int_{\mathbb{U}_2} f_2(t,X_{t-},Y_{t-},u) \, \widetilde{N}_2(\mathrm{d}t,\mathrm{d}u),\\ \mathrm{d}Y_t &= b_2(t,X_t,Y_t) \,\mathrm{d}t + \mathrm{d}W_t + \int_{\mathbb{U}_2} g(t,u) \, \widetilde{N}_2(\mathrm{d}t,\mathrm{d}u) \end{align}\] where, under the measure \(\mathbb{P}\), \(B\) and \(W\) are independent Brownian motions, \(N_1\) and \(N_2\) are random measures with compensators \(\nu_1(\mathrm{d}u) \,\mathrm{d}t\) and \(\lambda(t,X_{t-},u) \, \nu_2(\mathrm{d}u) \,\mathrm{d}t\) respectively, and we suppose that \(\log(\lambda(t,x,u)) = \kappa(t,x) g(t,u)\). Let us write \(Y^c\) and \(Y^d\) for the continuous and purely discontinuous local martingale parts of \(Y\) under \(\widetilde{\mathbb{P}}\), and let us adopt the assumptions of Theorem 6 with \(g = g_2 = g_3 = \gamma\), except with Assumption 7 replaced by the assumption that \(\mathcal{F}^Y_t = \mathcal{F}^Y_0 \vee \mathcal{F}^{Y^c}_t \vee \mathcal{F}^{Y^d}_t\) for every \(t \in [0,T]\).
Then, for every bounded measurable function \(F\) and every \(t \in [0,T]\), there exists a function \(\Theta^F_t\) on the space of càdlàg rough paths \(\mathscr{V}^p\), such that \[\pi_t(F) = \Theta^F_t\big(\mathbf{Y}^{c,d}\big)\] holds \(\mathbb{P}\)-almost surely, where we write \(\mathbf{Y}^{c,d}\) for the Itô rough path lift of the pair \((Y^c, Y^d)^\top\). Moreover, the function \(\Theta^F\) satisfies the same continuity properties as established for the corresponding function in Theorem 6.
Remark 24. We recall from Remark 19 that, when the function \(g\) is suitably regular, the condition \(\mathcal{F}^Y_t = \mathcal{F}^Y_0 \vee \mathcal{F}^{Y^c}_t \vee \mathcal{F}^{Y^d}_t\) is indeed satisfied. This includes, for instance, the case of additive Lévy noise, corresponding to the choice \(g(s,u) = u\) with \(\mathbb{U}_2 = \mathbb{R}^{d_Y} \setminus \{0\}\).
Remark 25. In light of Corollary 2, it may be tempting to ask whether it would be sufficient to lift each of \(Y^c\) and \(Y^d\) to rough paths \(\mathbf{Y}^c\) and \(\mathbf{Y}^d\) individually, and seek to represent the filter as a function of the pair \((\mathbf{Y}^c, \mathbf{Y}^d)\). However, even in the additive noise case, the Radon–Nikodym derivative includes the exponential of a stochastic integral of the form \[\int_0^t h(s,X_s,Y^c_s + Y^d_s)^\top \,\mathrm{d}Y^c_s,\] which in general is not continuous with respect to \((\mathbf{Y}^c, \mathbf{Y}^d)\), but only with respect to \(\mathbf{Y}^{c,d}\).
Remark 26. Let us recall the setting of Corollary 2, and let us further suppose that the function \(b_2\) is independent of \(y\), and that \(\kappa\) is a solution to the integral equation \[\kappa(t,x)^\top = b_2(t,x) + \int_{\mathbb{U}_2} g(t,u) \big( 1 - \exp \big(\kappa(t,x) g(t,u)\big) \big) \, \nu_2(\mathrm{d}u).\] In this case, we see from 30 that \(h^\top = \kappa\), and hence that the stochastic integral in 36 is given by \[\int_0^t H(s,X_{s-},Y_{s-}) \,\mathrm{d}G_s = \int_0^t \kappa(s,X_{s-}) \,\mathrm{d}Y_s.\] It follows that, in this (very) special case, it is not necessary to consider the continuous and jump parts of the observation separately, and there exists a robust representation of the conditional distribution which is a continuous function of \(\mathbf{Y}\), the Itô rough path lift of \(Y\).
Remark 27. Filtering assumes that the characteristics of the signal and noise are known to the observer, including in particular the compensator \(\nu_2(\mathrm{d}u) \,\mathrm{d}t\). It is therefore possible to reconstruct the purely discontinuous process \(Y^{d,n} := \int_0^\cdot \int_{\mathbb{U}_2} g(s,u) \mathbf{1}_{\{|g(s,u)| \geq \frac{1}{n}\}} \, \widetilde{N}(\mathrm{d}s,\mathrm{d}u)\) using discrete observations of \(Y\) path by path (see, e.g., part A or D of [61]). Hence, writing \(Y^{c,n} := Y - Y^{d,n}\), we can find consistent estimators of \(Y^d\) and \(Y^c\) which in turn can be arbitrarily well approximated using discrete observations of \(Y\). Therefore, Corollary 2 justifies the use of discrete approximations of \(Y^d\) and \(Y^c\), as long as there is a coherent way of lifting all these pairs of approximations to common rough paths which converge in the rough path topology to the lift of their corresponding limit.
For a normed vector space \((E,|\cdot|)\), a function \(f \colon E \to \mathbb{R}\), and any \(a, b, c, d \in E\), we write \[\label{eq:32124f12495144a44b44c44d} \begin{align} |f|_{1,\{a,b\}} &:= \sup_{\theta \in [0,1]} \big| f ( \theta a + (1 - \theta) b ) \big|,\\ |f|_{2,\{a,b,c,d\}} &:= \sup_{(\theta, \eta) \in [0,1]^2} \big| f \big( \eta ( \theta a + (1 - \theta) b ) + (1 - \eta) ( \theta c + (1 - \theta) d ) \big) \big|. \end{align}\tag{49}\]
Lemma 14. For some Banach space \((E,|\cdot|)\), let \(g \in C^2(E;\mathbb{R})\) and \(a, b, c, d \in E\). (We consider standard Fréchet differentiability here.) Then \[\label{elementary32estimate} \begin{align} &|g(a) - g(b) - g(c) + g(d)|\\ &\leq \big(|\mathrm{D}g|_{1,\{a,c\}} + |\mathrm{D}g|_{1,\{b,d\}} + |\mathrm{D}^2 g|_{2,\{a,b,c,d\}}\big) \big(|a-c| + |b-d|\big) \big( |c-d| \wedge 1 \big)\\ &\quad + |\mathrm{D}g|_{1,\{a,b\}} |a - b - c + d|. \end{align}\tag{50}\]
Proof. We first observe the straightforward bound \[\begin{align} |g(a) - g(b) - g(c) + g(d)| &\leq |g(a) - g(c)| + |g(b) - g(d)|\\ &\leq |\mathrm{D}g|_{1,\{a,c\}} |a - c| + |\mathrm{D}g|_{1,\{b,d\}} |b - d|. \end{align}\] We also have that \[\begin{align} &g(a) - g(b) - g(c) + g(d)\\ &= \int_0^1 \mathrm{D}g \big( b + \theta (a-b) \big) (a-b) \,\mathrm{d}\theta - \int_0^1 \mathrm{D}g \big( b + \theta (a-b) \big) (c-d) \,\mathrm{d}\theta\\ &\quad + \int_0^1 \mathrm{D}g \big( b + \theta (a-b) \big) (c-d) \,\mathrm{d}\theta - \int_0^1 \mathrm{D}g \big( d + \theta (c-d) \big) (c-d) \,\mathrm{d}\theta\\ &= \int_0^1 \mathrm{D}g \big( b + \theta (a-b) ) (a - b - c + d) \,\mathrm{d}\theta\\ &\quad + \int_0^1 \Big( \mathrm{D}g \big(b + \theta (a-b) \big) - \mathrm{D}g \big(d + \theta (c-d) \big) \Big) (c-d) \,\mathrm{d}\theta, \end{align}\] from which we infer that \[\begin{align} &|g(a) - g(b) - g(c) + g(d)|\\ &\leq |\mathrm{D}g|_{1,\{a,b\}} |a - b - c + d| + |\mathrm{D}^2 g|_{2,\{a,b,c,d\}} \big(|a - c| + |b - d|\big) |c - d|. \end{align}\] Combining the bounds above, we obtain 50 . ◻
Proof of Proposition 15. We note that, for any \(\lambda > 0\) and any \(m \in [1,\infty)\), we have \[\label{eq:32sup32V95u32has32exponential32moments} \Big\| \exp \Big( \lambda \sup_{u \in [0,T]} |V_u| \Big) \Big\|_{L^m} \leq \exp (\lambda \|V_0\|_{L^\infty}) \Big\| \mathbb{E}_0 \Big[ \exp \Big( m \lambda \sup_{u \in [0,T]} |\delta V_{0,u}| \Big) \Big] \Big\|_{L^\infty}^{\frac{1}{m}} < \infty,\tag{51}\] where, by Theorem 4, the right-hand side may be bounded by a constant which depends only on \(\lambda, m, p, \|V_0\|_{L^\infty}\) and \(\|V\|_{\mathrm{BMO}^{p\mathrm{-var}},[0,T]}\).
By the elementary estimate \(|\exp(x) - \exp(y)| \leq |x-y| (\exp(x) \vee \exp(y))\), and the version of Hölder’s inequality in 2 (with \(\ell = \frac{r}{q}\)), for any \((s,t) \in \Delta_{[0,T]}\), we have, for any \(q \in [1,\infty)\) and \(r \in [q,\infty)\), that \[\begin{align} &\|\delta \exp(V)_{s,t}\|_{q,r,s} \leq \Big\| \mathbb{E}_s \Big[ \exp \Big( q \sup_{u \in [s,T]} |V_u| \Big) |\delta V_{s,t}|^q \Big]^{\frac{1}{q}} \Big\|_{L^r}\\ &= \Big\| \mathbb{E}_s \Big[ \exp \Big( q \sup_{u \in [s,T]} |V_u| \Big) |\delta V_{s,t}|^q \Big] \Big\|_{L^{\frac{r}{q}}}^{\frac{1}{q}} \leq \Big\| \exp \Big( 2q \sup_{u \in [s,T]} |V_u| \Big) \Big\|_{L^{\frac{r}{q}}}^{\frac{1}{2q}} \|\delta V_{s,t}\|_{2q,2r,s}. \end{align}\] By 51 , this implies that \(\|\exp(V)\|_{p,q,r,[0,T]} \lesssim \|V\|_{p,2q,2r,[0,T]}\), which is finite by Corollary 1.
We now let \(1 \leq \tilde{q} \leq r < q < \infty\). Adopting the notation introduced in 49 , we let \(\Lambda = |\exp|_{2, \{V_t, \widetilde{V}_t, V_s, \widetilde{V}_s\}} + |\exp|_{1, \{V_t, \widetilde{V}_t\}} + |\exp|_{1, \{V_s, \widetilde{V}_s\}} + |\exp|_{1, \{V_t, V_s\}}\).
Applying Lemma 14, for any \((s,t) \in \Delta_{[0,T]}\), we have that \[\begin{align} &\big\|\delta \exp(V)_{s,t} - \delta \exp(\widetilde{V})_{s,t}\big\|_{\tilde{q},r,s}\\ &\leq \big\| \mathbb{E}_s \big[ \Lambda^{\tilde{q}} \big(|V_t - \widetilde{V}_t| + |V_s - \widetilde{V}_s|\big)^{\tilde{q}} |\delta \widetilde{V}_{s,t}|^{\tilde{q}} \big]^{\frac{1}{\tilde{q}}} \big\|_{L^r} + \big\| \mathbb{E}_s \big[ \Lambda^{\tilde{q}} |\delta V_{s,t} - \delta \widetilde{V}_{s,t}|^{\tilde{q}} \big]^{\frac{1}{\tilde{q}}} \big\|_{L^r}\\ &\lesssim \big\| \mathbb{E}_s \big[ \Lambda^{\tilde{q}} \big(|\delta V_{s,t} - \delta \widetilde{V}_{s,t}|\big)^{\tilde{q}} |\delta \widetilde{V}_{s,t}|^{\tilde{q}} \big]^{\frac{1}{\tilde{q}}} \big\|_{L^r} + \big\| \mathbb{E}_s \big[ \Lambda^{\tilde{q}} |V_s - \widetilde{V}_s|^{\tilde{q}} |\delta \widetilde{V}_{s,t}|^{\tilde{q}} \big]^{\frac{1}{\tilde{q}}} \big\|_{L^r}\\ &\quad + \big\| \mathbb{E}_s \big[ \Lambda^{\tilde{q}} |\delta V_{s,t} - \delta \widetilde{V}_{s,t}|^{\tilde{q}} \big]^{\frac{1}{\tilde{q}}} \big\|_{L^r} =: I_1 + I_2 + I_3. \end{align}\]
By Hölder’s inequality and 51 , it is straightforward to see that \(\|\Lambda\|_{L^m} < \infty\) for any \(m \in [1,\infty)\), and it is also clear that \(\|\delta \widetilde{V}_{s,t}\|_{L^m} < \infty\) for any \(m \in [1,\infty)\). We then have that \[I_1 \leq \big\| \mathbb{E}_s \big[ \Lambda^{\frac{\tilde{q}q}{q-\tilde{q}}} |\delta \widetilde{V}_{s,t}|^{\frac{\tilde{q}q}{q-\tilde{q}}} \big] \big\|_{L^{\frac{r}{\tilde{q}}}}^{\frac{q-\tilde{q}}{\tilde{q}q}} \|\delta V_{s,t} - \delta \widetilde{V}_{s,t}\|_{q,\frac{qr}{\tilde{q}},s} \lesssim \|\delta V_{s,t} - \delta \widetilde{V}_{s,t}\|_{q,\frac{qr}{\tilde{q}},s},\] and \(I_3\) may be treated similarly. Finally, using Hölder’s inequality again, we have that \[\begin{align} I_2 &\leq \|V_s - \widetilde{V}_s\|_{L^q} \big\| \mathbb{E}_s \big[ \Lambda^{\tilde{q}} |\delta \widetilde{V}_{s,t}|^{\tilde{q}} \big]^{\frac{1}{\tilde{q}}} \big\|_{L^{\frac{qr}{q-r}}} \leq \|V_s - \widetilde{V}_s\|_{L^q} \|\Lambda\|_{2\tilde{q},\frac{2qr}{q-r},s} \|\delta \widetilde{V}_{s,t}\|_{2\tilde{q},\frac{2qr}{q-r},s}\\ &\lesssim \big(\|V_0 - \widetilde{V}_0\|_{L^q} + \|V - \widetilde{V}\|_{p,q,\frac{qr}{\tilde{q}},[0,T]}\big) \|\delta \widetilde{V}_{s,t}\|_{2\tilde{q},\frac{2qr}{q-r},s}. \end{align}\] Combining the above, we obtain the estimate in ?? . ◻
The following result is a generalized conditional BDG inequality.
Lemma 15. Let \(q \in [2,\infty)\), and let \(M = (M_t)_{t \in [0,T]}\) be an \(L^q\)-integrable càdlàg real-valued local martingale with respect to a filtration \((\mathcal{F}_t)_{t \in [0,T]}\), and let \(\mathcal{G}\) be a sub-\(\sigma\)-algebra of \(\mathcal{F}_0\). There exist constants \(c_q, C_q\), which depend only on \(q\), such that \[\label{eq:32extended32conditional32BDG} c_q \mathbb{E}\Big[ \langle M \rangle_t^{\frac{q}{2}} \vee A^{(\frac{q}{2})}_t \,\Big|\, \mathcal{G}\Big] \leq \mathbb{E}\Big[ \sup_{s \in [0,t]} |M_s|^q \,\Big|\, \mathcal{G}\Big] \leq C_q \mathbb{E}\Big[ \langle M \rangle_t^{\frac{q}{2}} \vee A^{(\frac{q}{2})}_t \,\Big|\, \mathcal{G}\Big]\tag{52}\] holds almost surely for every \(t \in [0,T]\), where \(\langle M \rangle\) denotes the predictable quadratic variation of \(M\), and, in the notation of [62], we write \[A^{(\ell)} := \Pi^\ast_p \bigg( \sum_{s \leq \cdot} |\Delta M_s|^{2\ell} \bigg),\] where \(\Pi^\ast_p\) denotes the dual predictable projection.
Proof. This is a consequence of the extended BDG inequality in [62]. More precisely, for any \(G \in \mathcal{G}\), the process \((\widetilde{M}_t)_{t \in [0,T]} := (M_t \mathbf{1}_G)_{t \in [0,T]}\) is a local martingale, and \(\langle \widetilde{M}\rangle = \langle M \rangle \mathbf{1}_G\). Also, \(\widetilde{A}^{(\frac{q}{2})} := \Pi^\ast_p(\sum_{s \leq \cdot} |\Delta \widetilde{M}_s|^q) = \Pi^\ast_p(\sum_{s \leq \cdot} |\Delta M_s|^q \mathbf{1}_G) = A^{(\frac{q}{2})} \mathbf{1}_G\). Then, by the standard BDG inequality combined with [62], we have that \[\mathbb{E}\Big[ \Big( \sup_{s \in [0,t]} |M_s|^q \Big) \mathbf{1}_G \Big] = \mathbb{E}\Big[ \sup_{s \in [0,t]} |\widetilde{M}_s|^q \Big] \leq C_q \mathbb{E}\Big[ \langle \widetilde{M}\rangle_t^{\frac{q}{2}} \vee \widetilde{A}^{(\frac{q}{2})}_t \Big] = C_q \mathbb{E}\Big[ \Big( \langle M \rangle_t^{\frac{q}{2}} \vee A^{(\frac{q}{2})}_t \Big) \mathbf{1}_G \Big].\] Since this holds for all \(G \in \mathcal{G}\), we infer the second inequality in 52 , and the first inequality may be obtained similarly. ◻
Remark 28. Let \(N\) be an integer-valued random measure on \(\Omega \times [0,T] \times \mathbb{U}\) with compensator \(\nu\). Let \(\zeta \in G_{\mathrm{loc}}(N)\) be a real-valued function such that \(\mathbb{E}[\int_0^T \int_{\mathbb{U}} |\zeta(s,u)|^q \, \nu(\mathrm{d}s,\mathrm{d}u)] < \infty\) for some \(q \in [2,\infty)\), and let \(M = \int_0^\cdot \int_{\mathbb{U}} \zeta(s,u) \, \widetilde{N}(\mathrm{d}s,\mathrm{d}u)\). Then, with \(\beta\) and \(D\) defined as in 5 , we have by, e.g., [47], that \[\sum_{s \leq \cdot} |\Delta M_s|^q = \sum_{s \leq \cdot} |\zeta(s,\beta_s)|^q \mathbf{1}_D(s) = \int_0^\cdot \int_{\mathbb{U}} |\zeta(s,u)|^q \, N(\mathrm{d}s,\mathrm{d}u),\] and hence by, e.g., [44], that its dual predictable projection is given by \[\label{eq:32dual32predictable32projection32for32Poisson32martingale32integral} \Pi^\ast_p \bigg( \sum_{s \leq \cdot} |\Delta M_s|^q \bigg) = \int_0^\cdot \int_{\mathbb{U}} |\zeta(s,u)|^q \, \nu(\mathrm{d}s,\mathrm{d}u).\tag{53}\]
Lemma 16. Let \(N\) be an integer-valued random measure on \(\Omega \times [0,T] \times \mathbb{U}\), such that its compensator \(\nu\) satisfies \(\nu(\{t\} \times \mathbb{U}) = 0\) for every \(t \in [0,T]\), and write \(\widetilde{N}= N - \nu\) for the corresponding compensated random measure. Then the following hold.
Let \(q \in [2,\infty)\) and \(r \in [q,\infty]\). There exists a constant \(C\), which depends only on \(q\), such that, for any \(\zeta \in G_{\mathrm{loc}}(N)\) with \(\mathbb{E}[\int_0^T \int_{\mathbb{U}} |\zeta(s,u)|^q \, \nu(\mathrm{d}s,\mathrm{d}u)] < \infty\) and any \((s,h) \in \Delta_{[0,T]}\), \[\label{eq:32quadratic32variation32bound32for32compensated32poisson32martingale32integrals32using32BDG} \begin{align} \bigg\| &\sup_{v \in [s,h]} \bigg| \int_s^v \int_{\mathbb{U}} \zeta(t,u) \, \widetilde{N}(\mathrm{d}t,\mathrm{d}u) \bigg| \bigg\|_{q,r,s}\\ &\leq C \bigg\| \mathbb{E}_s \bigg[ \bigg( \int_s^h \int_{\mathbb{U}} |\zeta(t,u)|^2 \, \nu(\mathrm{d}t,\mathrm{d}u) \bigg)^{\frac{q}{2}} \vee \int_s^h \int_{\mathbb{U}} |\zeta(t,u)|^q \, \nu(\mathrm{d}t,\mathrm{d}u) \bigg]^{\frac{1}{q}} \bigg\|_{L^r}. \end{align}\tag{54}\]
Let \(\nu(\mathrm{d}t,\mathrm{d}u) = K(t,\mathrm{d}u) \,\mathrm{d}A_t\) as in 4 , and suppose that \(A \in V^{\frac{p}{2}} L^{\frac{q}{2},\frac{r}{2}} \cap V^{\frac{p}{q}} L^{1,\frac{r}{q}}\) for some \(2 \leq q \leq p < \infty\) and \(r \in [q,\infty]\), and let \(\zeta \in G_{\mathrm{loc}}(N)\) such that \[\bigg\| \sup_{t \in [0,T]} \int_{\mathbb{U}} \big( |\zeta(t,u)|^2 \vee |\zeta(t,u)|^q \big) \, K(t,\mathrm{d}u) \bigg\|_{L^\infty} \leq L\] for some finite constant \(L > 0\). Then there exists a constant \(C\), which depends only on \(q\) and \(L\), such that \[\bigg\| \int_0^\cdot \int_{\mathbb{U}} \zeta(t,u) \, \widetilde{N}(\mathrm{d}t,\mathrm{d}u) \bigg\|_{p,q,r,[0,T]} \leq C \Big( \|A\|_{\frac{p}{2},\frac{q}{2},\frac{r}{2},[0,T]}^{\frac{1}{2}} + \|A\|_{\frac{p}{q},1,\frac{r}{q},[0,T]}^\frac{1}{q} \Big).\]
Proof. By considering the integral \(\int_0^\cdot \int_{\mathbb{U}} \zeta(t,u) \, \widetilde{N}(\mathrm{d}t,\mathrm{d}u)\) componentwise, we may assume without loss of generality that \(\zeta\) is real-valued.
(i): By the conditional BDG inequality in Lemma 15, together with 53 , we have that \[\begin{align} &\bigg\| \mathbb{E}_s \bigg[ \sup_{v \in [s,h]} \bigg| \int_s^v \int_{\mathbb{U}} \zeta(t,u) \, \widetilde{N}(\mathrm{d}t,\mathrm{d}u) \bigg|^q \bigg]^{\frac{1}{q}} \bigg\|_{L^r}\\ &\lesssim \bigg\| \mathbb{E}_s \bigg[ \bigg\langle \int_s^\cdot \int_{\mathbb{U}} \zeta(t,u) \, \widetilde{N}(\mathrm{d}t,\mathrm{d}u) \bigg\rangle_{h}^{\frac{q}{2}} \vee \int_s^h \int_{\mathbb{U}} |\zeta(t,u)|^q \, \nu(\mathrm{d}t,\mathrm{d}u) \bigg]^{\frac{1}{q}} \bigg\|_{L^r}, \end{align}\] and the estimate in 54 then follows by [47].
(ii): Using the bound in 54 , for any \((s,v) \in \Delta_{[0,T]}\), we have that \[\begin{align} &\bigg\| \int_s^v \int_{\mathbb{U}} \zeta(t,u) \, \widetilde{N}(\mathrm{d}t,\mathrm{d}u) \bigg\|_{q,r,s}\\ &\lesssim \bigg\| \mathbb{E}_s \bigg[ \bigg( \int_s^v \int_{\mathbb{U}} |\zeta(t,u)|^2 \, K(t,\mathrm{d}u) \,\mathrm{d}A_t \bigg)^{\frac{q}{2}} \vee \int_s^v \int_{\mathbb{U}} |\zeta(t,u)|^q \, K(t,\mathrm{d}u) \,\mathrm{d}A_t \bigg]^{\frac{1}{q}} \bigg\|_{L^r}\\ &\leq L \Big( \|\delta A_{s,v}\|_{\frac{q}{2},\frac{r}{2},s}^{\frac{1}{2}} + \|\delta A_{s,v}\|_{1,\frac{r}{q},s}^{\frac{1}{q}} \Big) \leq L \Big( \|A\|_{\frac{p}{2},\frac{q}{2},\frac{r}{2},[s,v]}^{\frac{1}{2}} + \|A\|_{\frac{p}{q},1,\frac{r}{q},[s,v]}^{\frac{1}{q}} \Big) \end{align}\] and, since \(q \leq p\), the desired estimate then follows by superadditivity. ◻
Proof of Proposition 7. Let \(t \in [0,T]\). By, e.g., [63], there exists a sequence of partitions \((\mathcal{P}^k)_{k \in \mathbb{N}}\) of the interval \([0,t]\) with vanishing mesh size, such that the limit \[\label{eq:32lim32Yu32delta32Xuv32int32Ys-32dXs} \lim_{k \to \infty} \sum_{[u,v] \in \mathcal{P}^k} Y_u \delta X_{u,v} = \int_0^t Y_{s-} \,\mathrm{d}X_s\tag{55}\] holds \((\mathbb{P}\otimes \bar{\mathbb{P}})\)-almost surely.
By [64], we have that \(\sum_{[u,v] \in \mathcal{P}^k} Y'_{u-} \mathbb{X}_{u,v} \to 0\) as \(k \to \infty\) as a limit in probability with respect to \(\mathbb{P}\otimes \bar{\mathbb{P}}\), and by switching to a subsequence if necessary, we may assume that the limit \[\label{eq:32lim32sum32Y39u-32Xuv3261320} \lim_{k \to \infty} \sum_{[u,v] \in \mathcal{P}^k} Y'_{u-} \mathbb{X}_{u,v} = 0\tag{56}\] holds \((\mathbb{P}\otimes \bar{\mathbb{P}})\)-almost surely.
We have by assumption that \((Y(\omega, \cdot),Y'(\omega, \cdot)) \in \mathcal{V}_{X(\omega)}^{p,q,r,\bar{\Omega}}\) for \(\mathbb{P}\)-almost every \(\omega \in \Omega\). For any such \(\omega \in \Omega\), letting \(\Xi_{s,t} = \Delta Y'_s(\omega, \cdot) \mathbb{X}_{s,t}(\omega)\), we have that \[\|\Xi_{s,t}\|_{L^q(\bar{\Omega})} \leq \delta(s)^{\frac{1}{p}} w(s,t)^{\frac{2}{p}},\] where \(\delta(s) := \|\Delta Y'_s(\omega, \cdot)\|_{L^q(\bar{\Omega})}^p\) and \(w(s,t) := \|\mathbb{X}(\omega)\|_{\frac{p}{2},[s,t]}^{\frac{p}{2}}\). It then follows from [46] that \(\sum_{[u,v] \in \mathcal{P}^k} \Delta Y'_u(\omega, \cdot) \mathbb{X}_{u,v}(\omega) \to 0\) as \(k \to \infty\) in probability with respect to \(\bar{\mathbb{P}}\). Since this convergence holds for \(\mathbb{P}\)-almost every \(\omega \in \Omega\), it follows that \(\sum_{[u,v] \in \mathcal{P}^k} \Delta Y'_u \mathbb{X}_{u,v} \to 0\) as \(k \to \infty\) in probability with respect to \(\mathbb{P}\otimes \bar{\mathbb{P}}\). By switching to a further subsequence if necessary, we may then assume that the limit \[\label{eq:32lim32Delta32Y3995u32Xuv3261320} \lim_{k \to \infty} \sum_{[u,v] \in \mathcal{P}^k} \Delta Y'_u \mathbb{X}_{u,v} = 0\tag{57}\] holds \(\mathbb{P}\otimes \bar{\mathbb{P}}\)-almost surely, and since \(Y'_u = Y'_{u-} + \Delta Y'_u\), combining the limits in 56 and 57 , we have that \[\label{eq:32lim32sum32Y39u32Xuv3261320} \lim_{k \to \infty} \sum_{[u,v] \in \mathcal{P}^k} Y'_u \mathbb{X}_{u,v} = 0\tag{58}\] also holds \(\mathbb{P}\otimes \bar{\mathbb{P}}\)-almost surely.
Combining 55 and 58 , we have in particular, by Fubini’s theorem (e.g., [65]), that for \(\mathbb{P}\)-almost every \(\omega \in \Omega\), the limit \[\lim_{k \to \infty} \sum_{[u,v] \in \mathcal{P}^k} Y_u(\omega, \cdot) \delta X_{u,v}(\omega) + Y'_u(\omega, \cdot) \mathbb{X}_{u,v}(\omega) = \bigg( \int_0^t Y_{s-} \,\mathrm{d}X_s \bigg) (\omega, \cdot)\] holds \(\bar{\mathbb{P}}\)-almost surely. However, by the definition of the rough stochastic integral (see Lemma 1), we also have that, for \(\mathbb{P}\)-almost every \(\omega \in \Omega\), \[\lim_{k \to \infty} \sum_{[u,v] \in \mathcal{P}^k} Y_u(\omega, \cdot) \delta X_{u,v}(\omega) + Y'_u(\omega, \cdot) \mathbb{X}_{u,v}(\omega) = \int_0^t Y_s(\omega, \cdot) \,\mathrm{d}\mathbf{X}_s(\omega)\] exists as a limit in probability with respect to \(\bar{\mathbb{P}}\). By the \(\bar{\mathbb{P}}\)-almost sure uniqueness of limits, for \(\mathbb{P}\)-almost every \(\omega \in \Omega\), we have that \[\bigg( \int_0^t Y_{u-} \,\mathrm{d}X_u \bigg)(\omega, \cdot) = \int_0^t Y_u(\omega, \cdot) \,\mathrm{d}\mathbf{X}_u(\omega),\] \(\bar{\mathbb{P}}\)-almost surely. Since this holds for every \(t \in [0,T]\), and both Itô and rough stochastic integrals have almost surely càdlàg sample paths, we deduce the result. ◻
::: {#lemma: |bX|_p,[0,T] leq N_alpha + jumps .lemma} Lemma 17. Let \(p \in [2,3)\) and let \(\mathbf{X}\in \mathscr{V}^p\) be a rough path. For any \(\alpha > 0\), we have that \[\label{eq:32bound32on32rough32path32norm32in32terms32of32N95alpha32and32jumps} \|\mathbf{X}\|_{p,[0,T]} \leq C \Big( N_{\alpha,[0,T]}\big(\|\mathbf{X}\|_{p,[\cdot,\cdot)}^p\big) + 1 \Big)^{3} \Big( 1 + \sup_{t \in (0,T]} |\Delta \mathbf{X}_t| \Big)^{2},\tag{59}\] where the constant \(C\) depends only on \(p\) and \(\alpha\). :::
Proof. Let us write \(N = N_{\alpha,[0,T]}(\|\mathbf{X}\|_{p,[\cdot,\cdot)}^p)\) and \(\{t_i\}_{i=0}^{N+1} = \{t_i(\alpha)\}_{i=0}^{N+1}\) for the partition defined in 25 , with \(t_{N+1} = T\). Let \(\mathcal{P}\) be an arbitrary partition of the interval \([0,T]\), and let \(\bar{\mathcal{P}} = \mathcal{P}\cup \{t_i\}_{i=0}^{N+1}\). Similarly to the proof of [57], we bound \[\begin{align} \sum_{[u,v] \in \mathcal{P}} |\delta X_{u,v}|^p &\leq (N+1)^{p-1} \sum_{[s,t] \in \bar{\mathcal{P}}} |\delta X_{s,t}|^p \leq (N+1)^{p-1} \sum_{i=0}^N \|X\|_{p,[t_i,t_{i+1}]}^p\\ &\leq (N+1)^{p-1} \sum_{i=0}^{N-1} \big( \|X\|_{p,[t_i,t_{i+1})} + |\Delta X_{t_{i+1}}| \big)^p \lesssim (N+1)^p \Big(\alpha + \sup_{t \in (0,T]} |\Delta X_t|^p \Big), \end{align}\] where we used the fact that \(\|X\|_{p,[t_i,t_{i+1}]} \leq \|X\|_{p,[t_i,t_{i+1})} + |\Delta X_{t_{i+1}}|\) (see [66]), which implies that \[\|X\|_{p,[0,T]} \lesssim \Big( N_{\alpha,[0,T]}\big(\|\mathbf{X}\|_{p,[\cdot,\cdot)}^p\big) + 1 \Big) \Big( 1 + \sup_{t \in (0,T]} |\Delta \mathbf{X}_t| \Big).\] A similar argument can then be used to bound \(\|\mathbb{X}\|_{\frac{p}{2},[0,T]}\). This is more tedious, since \(\mathbb{X}\) is not additive, and using Chen’s relation introduces additional terms. In particular, we use the inequality \(\|\mathbb{X}\|_{\frac{p}{2},[s,t]} \lesssim \|\mathbb{X}\|_{\frac{p}{2},[s,t)} + |\Delta \mathbb{X}_t| + \|X\|_{p,[s,t)}^2 + |\Delta X_t|^2\), which follows from a straightforward adaptation of the proof of [66]. After some calculation, which we omit for brevity, we arrive at the estimate in 59 . ◻
Proof of Theorem 2. Let us first consider the latter estimate. We note that when \(q = p\) the claim follows easily by the linearity of expectation.
When \(q < p\) the result essentially follows from Minkowski’s integral inequality (see, e.g., [65]), but for completeness we spell out the argument here. We first fix a (deterministic) partition \(\mathcal{P}= \{t_i\}_{i=0}^N\) of \([0,T]\). Letting \(f(i,\omega) := |\delta Y_{t_i,t_{i+1}}(\omega)|\), we have that the function \(f \colon \{0, 1, \ldots, N-1\} \times \Omega \to [0,\infty)\) is jointly measurable. Let us also write \(\mu\) for the counting measure on \(\{0, 1, \ldots, N-1\}\). By Minkowski’s integral inequality, we then have that \[\begin{align} \bigg( &\sum_{i=0}^{N-1} \|\delta Y_{t_i,t_{i+1}}\|_{L^q}^p \bigg)^{\frac{1}{p}} = \bigg( \int_{\{0, \ldots, N-1\}} \bigg( \int_{\Omega} |f(i,\omega)|^q \,\mathrm{d}\mathbb{P}(\omega) \bigg)^{\frac{p}{q}} \,\mathrm{d}\mu(i) \bigg)^{\frac{1}{p}}\\ &\leq \bigg( \int_{\Omega} \bigg( \int_{\{0, \ldots, N-1\}} |f(i,\omega)|^p \,\mathrm{d}\mu(i) \bigg)^{\frac{q}{p}} \,\mathrm{d}\mathbb{P}(\omega) \bigg)^{\frac{1}{q}} = \bigg\| \bigg( \sum_{i=0}^{N-1} |\delta Y_{t_i,t_{i+1}}|^p \bigg)^{\frac{1}{p}} \bigg\|_{L^q}, \end{align}\] which implies the desired estimate.
Let us now take \(p < q\) and consider the first estimate. Let \(w\) be the control function given by \(w(s,t) = \|Y\|_{p,q,[s,t]}^p\) for \((s,t) \in \Delta_{[0,T]}\). We may assume without loss of generality that \(\tilde{p}\leq q\), and that \(\Delta Y_T = 0\).
Let \(d^n_i\) for \(n \in \mathbb{N}\) and \(i = 0, 1, \ldots, 2^n\) be the \(w(\cdot, \cdot-)\)-midpoints of the interval \([0,T]\), in the sense of [67]. For convenience, let us refer to \(D_n = \{[d^n_i,d^n_{i+1}] : i = 0, 1, \ldots, 2^n-1\}\) as the \(w\)-dyadic partition at level \(n\), and write \(\mathcal{D}= \cup_{n \in \mathbb{N}} D_n\) for the set of all \(w\)-dyadic points.
We may assume without loss of generality that the mesh size \(|D_n| \to 0\) as \(n \to \infty\). (Indeed, if not, we may simply add the function \((s,t) \mapsto \varepsilon(t - s)\) to the control \(w\), which will ensure that \(|D_n| \to 0\), and then take \(\varepsilon\to 0\) at the end of the proof.) This means in particular that \(\mathcal{D}\) is dense in \([0,T]\).
Since \(p < \tilde{p}\), we have, by a standard property of such midpoints (see, e.g., [22]), that for every \(n \in \mathbb{N}\), \[\label{eq:32midpoints32bound32for32sum32alt46} \sum_{i=0}^{2^n-1} w(d^n_i,d^n_{i+1}-)^{\frac{\tilde{p}}{p}} \leq 2^{-(\frac{\tilde{p}}{p} - 1) n} w(0,T-)^{\frac{\tilde{p}}{p}}.\tag{60}\]
Given arbitrary points \(s, t \in \mathcal{D}\) with \(s < t\), the interval \([s,t]\) can be expressed as the finite union of essentially disjoint intervals of the form \([d^n_i,d^n_{i+1}]\), where no three intervals are in the same level \(n\). In other words, for each \(n \in \mathbb{N}\), there exists an index set \(I_n^{s,t} \subset \{0, 1, \ldots, 2^n-1\}\), with \(|I_n^{s,t}| \in \{0, 1, 2\}\) and \(I_n^{s,t} = \emptyset\) for sufficiently large \(n\), such that \[[s,t] = \bigcup_{n=1}^\infty \bigcup_{i \in I_n^{s,t}} [d^n_i,d^n_{i+1}],\] where all the intervals are disjoint except for their endpoints. In particular, we can write \[\delta Y_{s,t} = \sum_{n=1}^\infty \sum_{i \in I_n^{s,t}} \delta Y_{d^n_i,d^n_{i+1}}.\]
Let \(\gamma > \tilde{p}- 1\). Using the fact that \((\sum_{n=1}^\infty a_n)^{\tilde{p}} \leq C \sum_{n=1}^\infty n^\gamma a_n^{\tilde{p}}\) for any sequence \((a_n)_{n \in \mathbb{N}}\) of non-negative numbers, where the constant \(C\) depends only on \(\tilde{p}\) and \(\gamma\) (see the proof of [68]), we then have that \[|\delta Y_{s,t}|^{\tilde{p}} \leq C \sum_{n=1}^\infty n^\gamma \bigg( \sum_{i \in I^{s,t}_n} |\delta Y_{d^n_i,d^n_{i+1}}| \bigg)^{\tilde{p}} \leq C 2^{\tilde{p}-1} \sum_{n=1}^\infty n^\gamma \sum_{i \in I^{s,t}_n} |\delta Y_{d^n_i,d^n_{i+1}}|^{\tilde{p}}.\]
Let us now take a partition \(\{0 = t_0 < t_1 < \cdots < t_m = T\} \subset \mathcal{D}\) of the interval \([0,T]\). We can apply the procedure above to each of the intervals \([t_j,t_{j+1}]\) in this partition. Noting that each dyadic point \(d^n_{i+1} \in \mathcal{D}\) appears at most once as the right-endpoint of one of the considered intervals (i.e., we need sum each jump \(\Delta Y_{d^n_{i+1}}\) at most once), we see that \[\label{eq:32pathwise32bound32for32sums32over32partitions32alt46} \begin{align} \sum_{j=0}^{m-1} |\delta Y_{t_j,t_{j+1}}|^{\tilde{p}} &\leq C 2^{2(\tilde{p}-1)} \bigg( \sum_{n=1}^\infty n^\gamma \sum_{i=0}^{2^n-1} |\delta Y_{d^n_i,d^n_{i+1}-}|^{\tilde{p}} + \sum_{n =1}^{\infty} n^{\gamma}\sum_{i=0}^{2^{n-1}-1} |\Delta Y_{d^n_{2i+1}}|^{\tilde{p}} \bigg)\\ &=: C 2^{2(\tilde{p}-1)} \big( Z_1^{\tilde{p}} + Z_2^{\tilde{p}} \big). \end{align}\tag{61}\] Using the bound in 60 , we have that \[\sum_{i=0}^{2^n-1} \|\delta Y_{d^n_i,d^n_{i+1}-}\|_{L^q}^{\tilde{p}} \leq \sum_{i=0}^{2^n-1} w(d^n_i,d^n_{i+1}-)^{\frac{\tilde{p}}{p}} \leq 2^{-(\frac{\tilde{p}}{p} - 1) n} w(0,T)^{\frac{\tilde{p}}{p}} = 2^{-(\frac{\tilde{p}}{p} - 1) n} \|Y\|_{p,q,[0,T]}^{\tilde{p}},\] and hence, since \(p < \tilde{p}\leq q\), we have that \[\label{eq:32Z95132estimate} \begin{align} \| Z_1 \|_{L^q}^{\tilde{p}} &= \bigg\| \sum_{n=1}^\infty n^\gamma \sum_{i=0}^{2^n-1} |\delta Y_{d^n_i,d^n_{i+1}-}|^{\tilde{p}} \bigg\|_{L^{\frac{q}{\tilde{p}}}} \leq \sum_{n=1}^\infty n^\gamma \sum_{i=0}^{2^n-1} \big\| |\delta Y_{d^n_i,d^n_{i+1}-}|^{\tilde{p}} \big\|_{L^{\frac{q}{\tilde{p}}}}\\ &= \sum_{n=1}^\infty n^\gamma \sum_{i=0}^{2^n-1} \| \delta Y_{d^n_i,d^n_{i+1}-} \|_{L^q}^{\tilde{p}} \leq \|Y\|_{p,q,[0,T]}^{\tilde{p}} \sum_{n=1}^\infty n^\gamma 2^{-(\frac{\tilde{p}}{p} - 1) n} < \infty. \end{align}\tag{62}\]
Noting that \(d^n_{2i+1} \in (d^{n-1}_i, d^{n-1}_{i+1})\) for each \(i \in \{0, \ldots, 2^{n-1}-1\}\), we have that \[\|\Delta Y_{d^n_{2i+1}}\|^{\tilde{p}}_{L^q} = \lim_{s \nearrow d^n_{2i+1}} \|\delta Y_{s,d^n_{2i+1}}\|_{L^q}^{\tilde{p}} \leq w(d^{n-1}_i, d^{n-1}_{i+1}-)^{\frac{\tilde{p}}{p}},\] and hence that \[\label{eq:32Z95232estimate} \begin{align} \| Z_2 \|_{L^q}^{\tilde{p}} &\leq \sum_{n=1}^\infty n^\gamma \sum_{i=0}^{2^{n-1}-1} \| \Delta Y_{d^n_{2i+1}} \|_{L^q}^{\tilde{p}} \leq \sum_{n=1}^\infty n^\gamma \sum_{i=0}^{2^{n-1}-1} w(d^{n-1}_i, d^{n-1}_{i+1}-)^{\frac{\tilde{p}}{p}}\\ &\leq \sum_{n=1}^\infty n^\gamma 2^{-(\frac{\tilde{p}}{p} - 1) (n-1)} w(0,T)^{\frac{\tilde{p}}{p}} = \|Y\|_{p,q,[0,T]}^{\tilde{p}} \sum_{n=1}^\infty n^\gamma 2^{-(\frac{\tilde{p}}{p} - 1) (n-1)} < \infty. \end{align}\tag{63}\] We thus have in particular that the right-hand side of 61 is almost surely finite.
It follows that almost every sample path of \(Y\) has finite \(\tilde{p}\)-variation on the \(w\)-dyadic times \(\mathcal{D}\). In particular, these sample paths are regulated (in the sense that their left and right-limits exist at every point). We also recall that the dyadic times \(\mathcal{D}\) are dense in \([0,T]\). Thus, for every non-dyadic time \(t \in (0,T) \setminus \mathcal{D}\), the limit \(\widetilde{Y}_t := \lim_{k \to \infty} Y_{t_k}\) exists almost surely whenever \((t_k)_{k \in \mathbb{N}} \subset \mathcal{D}\) is a decreasing sequence of dyadic times with \(t_k \searrow t\), and the limit does not depend on the choice of the sequence \((t_k)_{k \in \mathbb{N}}\). Setting \(\widetilde{Y}_t = Y_t\) for all dyadic times \(t \in \mathcal{D}\), we obtain a process \(\widetilde{Y}\) on \([0,T]\).
It is straightforward to see that, by construction, the sample paths of \(\widetilde{Y}\) are almost surely càdlàg and have finite \(\tilde{p}\)-variation on \([0,T]\). Since \(Y_{t_k} \to \widetilde{Y}_t\) almost surely, and \(Y_{t_k} \to Y_t\) in probability as \(k \to \infty\), we have that \(\widetilde{Y}_t = Y_t\) almost surely for every \(t \in [0,T]\), so that \(\widetilde{Y}\) is indeed a modification of \(Y\).
Finally, by combining 61 with the bounds in 62 and 63 , we deduce the estimate in 15 . ◻
Example 3. Let \((\Omega,\mathcal{F},\mathbb{P})\) be a probability space, and let \(p, \tilde{p}, q \geq 1\). For each \(k \geq 1\), let \((E_{k,i})_{i = 1, \ldots, k} \subset \mathcal{F}\) be a collection of events which form a partition of the sample space \(\Omega\), such that \(\mathbb{P}(E_{k,i}) = \frac{1}{k}\) for each \(i = 1, \ldots, k\). We define random variables \((\zeta_{k,i})_{i = 1, \ldots, k}\) by \[\zeta_{k,i} = k^{-\frac{1}{\tilde{p}}} \mathbf{1}_{E_{k,i}}\] for each \(i = 1, \ldots, k\), and let \(Z\) be the sequence of random variables given by \[Z := (0, \zeta_{1,1}, 0, \zeta_{2,1}, 0, \zeta_{2,2}, 0, \zeta_{3,1}, 0, \zeta_{3,2}, 0, \zeta_{3,3}, 0, \zeta_{4,1}, \ldots).\] Then, for every \(\omega \in \Omega\), we have that \[\|Z(\omega)\|_{\tilde{p}}^{\tilde{p}} = 2 \sum_{k=1}^\infty k^{-1} = \infty.\]
For any \(k \leq m\), any \(i = 1, \ldots, k\) and any \(j = 1, \ldots, m\), noting that \(\mathbb{P}(\zeta_{k,i} - \zeta_{m,j} \neq 0) \leq \frac{1}{k} + \frac{1}{m} \leq \frac{2}{k}\), we see that \[\|\zeta_{k,i} - \zeta_{m,j}\|_{L^q} \leq k^{-\frac{1}{\tilde{p}}} \Big(\frac{2}{k}\Big)^{\frac{1}{q}} = 2^{\frac{1}{q}} k^{-\frac{1}{q} - \frac{1}{\tilde{p}}}.\] Hence, we can bound \[\|Z\|_{p,q}^p \leq \sum_{k=1}^\infty 2k \cdot 2^{\frac{p}{q}} k^{-\frac{p}{q} - \frac{p}{\tilde{p}}} \leq 2^{1+\frac{p}{q}} \sum_{k=1}^\infty k^{1 - \frac{p}{q} - \frac{p}{\tilde{p}}}.\] The sum above is finite when \[\label{eq:32pqtp32condition} \frac{1}{q} + \frac{1}{\tilde{p}} > \frac{2}{p}.\tag{64}\] Thus, whenever the condition in 64 is satisfied, we conclude that \[\|Z\|_{p,q} < \infty \qquad \text{but} \qquad \|Z(\omega)\|_{\tilde{p}} = \infty \text{~~for every~~} \omega \in \Omega.\] In particular, if \(p > q\) then we can take \(\tilde{p}> p\) above, which implies that the estimate in 15 does not hold in general without the assumption that \(p < q\).
Moreover, if \(p < q\) then 64 requires that \(\tilde{p}< p\), but if \(p \approx q\) then we can also take \(\tilde{p}\approx p\), indicating that in general 15 also cannot hold for \(\tilde{p}< p\).
Department of Mathematical Sciences, Durham University, andrew.l.allan@durham.ac.uk↩︎
Department of Mathematical Sciences, Durham University, jost.pieper@durham.ac.uk↩︎
Department of Mathematics, ETH Zürich, josef.teichmann@math.ethz.ch↩︎
In [13] this argument was omitted, but it has become apparent that this is a substantially more delicate task than the authors of [13] suggest. Indeed, this problem appears to have been resolved satisfactorily for the first time only very recently in [23]. See Section 3 below for an alternative argument.↩︎
We choose this slightly non-standard definition of the rough path norm for the convenient property that the map \((s,t) \mapsto \|\mathbf{X}\|_{p,[s,t]}^p\) is a control.↩︎
To see this, one can simply take a partition \(\mathcal{P}= \{t_i\}\) of \([0,T]\) such that \(\|\mathbf{X}\|_{p,[t_i,t_{i+1})}\) is small for each \(i\), which allows one to make the uniform distance \(\|\mathbf{X}^\mathcal{P}- \mathbf{X}\|_{\infty,[0,T]}\) arbitrarily small, and then use interpolation of \(p\)-variation norms to obtain a bound on \(\|\mathbf{X}^\mathcal{P}- \mathbf{X}\|_{p',[0,T]}\) for \(p' > p\).↩︎
Although the result in [50] includes certain discontinuous paths, it does not appear to apply to general càdlàg paths.↩︎
We implicitly assume that \(\hat{\mathcal{F}}_t\) has been suitably augmented, so that \((\hat{\mathcal{F}}_t)_{t \in [0,T]}\) satisfies the usual conditions.↩︎
We may ignore the dependence of the constant \(C_1\) (as well as the constant \(C_2\) below) on \(\|X(\omega)\|_{p,[\tau_i(\omega),\tau_{i+1}(\omega)]}\) here, thanks to 21 and the fact that \(\alpha \leq 1\).↩︎
The extra term in the metric in 18 , compared to the metric used in the proof of [22], results in a slightly different constant \(C_2\) here, but this is immaterial.↩︎