June 01, 2026
We introduce the class of Special–Dirichlet processes, consisting of càdlàg adapted processes admitting a decomposition \[X=M+\Gamma,\] where \(M\) is a local martingale and \(\Gamma\) is an adapted càdlàg process with vanishing continuous quadratic variation whose jumps are predictable and \(\mathcal{F}_{s-}\)-measurable.
This class arises naturally from transformations of special semimartingales. Classical results imply that sufficiently regular functions of special semimartingales belong to the broad class of Dirichlet processes. We show that such transformed processes possess substantially more structure: they admit a canonical decomposition in which the predictable jump component is explicitly separated from the martingale component. This yields a refinement of the traditional classification, which previously identified these processes only as Dirichlet processes.
We establish uniqueness of the decomposition and prove that the class is stable under a large family of nonsmooth transformations, including primitives of locally bounded functions with at most countably many discontinuities. An explicit Itô-type decomposition is obtained in terms of the martingale jump measure and its compensator.
Finally, we investigate stability properties of the canonical decomposition. Under convergence in quadratic variation and Skorokhod \(J_1\)-convergence, we prove stability of both the martingale and singular components after transformation. The proof relies on a threshold isolation principle for jump structures, allowing large jumps to be separated from small-jump contributions and yielding convergence of the transformed decompositions.
The class of Dirichlet processes was introduced in [@Fol] as an extension of the class of continuous semimartingales. In the original framework, a Dirichlet process admits a decomposition \[X=M+A,\] where \(M\) is a martingale and \(A\) is a continuous process of zero energy.
Subsequent developments led to several extensions of the original Dirichlet framework. In [@NonCont], the Dirichlet framework was extended beyond the continuous semimartingale setting by considering decompositions \[X=Z+C,\] where \(Z\) is a semimartingale and \(C\) is continuous with \([C]=0\). Stability questions under quadratic variation convergence in this setting were later studied in [@Collectanea]. A different direction was pursued in [@WeakDir], which introduced weak Dirichlet processes. More recently, [@Low] considered processes of the form \[X=Z+C,\] where \(Z\) is a semimartingale and \(C\) is a càdlàg adapted process satisfying \[[C]^c=0.\]
This setting provides the natural ambient class for the present work. In [@Low], Itô-type decomposition results were obtained for locally Lipschitz functions of semimartingales and, more generally, for Dirichlet processes under suitable non-charging conditions on the continuous quadratic variation.
We adopt the notion of Dirichlet processes in the sense of Lowther, namely processes of the form \[X = Z + C,\] where \(Z\) is a semimartingale and \(C\) is a càdlàg adapted process satisfying \[[C]^c = 0.\] This definition allows for jump components in \(C\) and is therefore suited to the present setting.
We also recall the classical (CMS/Föllmer-type) definition of Dirichlet processes, where the residual component is required to be continuous with zero quadratic variation. In particular, this excludes jump contributions in the residual.
Finally, we consider weak Dirichlet processes in the sense of Russo et al., which are defined by a decomposition \[X = M + A,\] where \(M\) is a local martingale and \(A\) is orthogonal to all continuous local martingales.
The present work originates from the observation that the usual Dirichlet classification is too coarse for many processes arising from transformation theory. In particular, transformations of special semimartingales are not merely Dirichlet processes in the generalized sense of [@Low]. They possess a finer structure: after transformation, the martingale and predictable jump contributions can be separated in a canonical way.
A key motivation for introducing the Special–Dirichlet class is that it arises naturally as the recipient of such transformation results. Indeed, we show that if \(Z\) is a special semimartingale and \(f\) belongs to a broad nonsmooth class, then \(f(Z)\) is a Special–Dirichlet process. Thus the traditional conclusion that \(f(Z)\) is Dirichlet can be sharpened: the transformed process admits a canonical decomposition in which the local martingale part and the predictable singular component are explicitly separated.
This is the main motivation for introducing the class of Special–Dirichlet processes. These are processes admitting a decomposition \[X=M+\Gamma,\] where \(M\) is a local martingale and \(\Gamma\) is a càdlàg adapted process with vanishing continuous quadratic variation whose jumps are carried by a thin predictable set and satisfy \[\Delta\Gamma_s\in\mathcal{F}_{s-}.\] The definition is modeled on the role of special semimartingales within the class of semimartingales, but it is designed for the Dirichlet setting: the singular component is allowed to have jumps, while its jump structure is required to be predictable.
With these conventions, the following inclusions hold: \[\boxed{ \text{Special--Dirichlet} \;\subset\; \text{Dirichlet (Lowther)} \;\subset\; \text{weak Dirichlet}. }\]
The first inclusion follows from the fact that in our framework the residual component \(\Gamma\) satisfies \([\Gamma]^c=0\), and hence fits into the Lowther definition with \(Z=M\) and \(C=\Gamma\). However, since \(\Gamma\) may exhibit jumps, our class is in general not contained in the classical (CMS) Dirichlet class, where the residual component is required to be continuous.
Dirichlet processes in the sense of [@Low] do not admit a canonical decomposition in general, since the splitting into semimartingale and zero–continuous–quadratic–variation parts is not unique. By contrast, weak Dirichlet processes admit a unique decomposition under the orthogonality condition.
The uniqueness result therefore shows that the Special–Dirichlet condition identifies a canonical substructure inside the general Lowther framework: the decomposition is unique, while the singular component retains an explicit interpretation in terms of predictable jumps and vanishing continuous quadratic variation.
The class also has strong stability properties. We prove stability of the canonical decomposition under convergence in quadratic variation, and we establish stability under transformations. In the latter result, simultaneous convergence in quadratic variation and in the Skorokhod \(J_1\)-topology yields convergence of both the martingale and singular components after transformation. The proof relies on a threshold isolation principle for jump structures, allowing large jumps to be separated from small-jump contributions.
The article is organized as follows. Section 2 introduces notation and preliminary material. Section 3 develops the basic theory of Special–Dirichlet processes and establishes uniqueness of the canonical decomposition. Section 4 proves stability of the decomposition itself. Section 5 establishes closure under nonsmooth transformations and proves stability of the transformed decompositions.
We assume that all processes are defined on a common filtered probability space \[(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge0},\mathbb{P}),\] and that all processes are adapted to a filtration satisfying the usual conditions.
The term refining sequence will refer to a sequence \[D_n=\{\tau_0^n,\tau_1^n,\ldots,\tau_{m_n}^n\}\] of finite increasing families of stopping times satisfying \[0=\tau_0^n\le\tau_1^n\le\cdots\le\tau_{m_n}^n=t,\] and \[|D_n| := \max_{0\le i<m_n} (\tau_{i+1}^n-\tau_i^n) \xrightarrow{\mathbb{P}}0.\]
We say that a càdlàg process \(X\) admits a quadratic variation \([X]\) if there exists an increasing continuous process \([X]^c\) such that \[\begin{align} \label{quad} [X]_s = [X]^c_s + \sum_{u\le s}(\Delta X_u)^2, \qquad 0\le s\le t, \end{align}\tag{1}\] and there exists a refining sequence \(\{D_n\}_n\) such that \[(S_n(X))_s := \sum_{i=0}^{m_n-1} \Bigl( X_{\tau_{i+1}^n\wedge s} - X_{\tau_i^n\wedge s} \Bigr)^2\] satisfies \[\begin{align} \label{partialsums} (S_n(X))_s \xrightarrow{\mathbb{P}} [X]_s \qquad\text{as }n\to\infty, \end{align}\tag{2}\] for every \(0\le s\le t\).
We say that two càdlàg processes \(X\) and \(Y\) admit a covariation \([X,Y]\) if there exists a continuous finite variation process \([X,Y]^c\) such that \[[X,Y]_s = [X,Y]^c_s + \sum_{u\le s}\Delta X_u\,\Delta Y_u, \qquad 0\le s\le t,\] and there exists a refining sequence \(\{D_n\}_n\) such that \[S_n(X,Y)_s := \sum_{i=0}^{m_n-1} \Bigl( X_{\tau_{i+1}^n\wedge s} - X_{\tau_i^n\wedge s} \Bigr) \Bigl( Y_{\tau_{i+1}^n\wedge s} - Y_{\tau_i^n\wedge s} \Bigr)\] satisfies \[S_n(X,Y)_s \xrightarrow{\mathbb{P}} [X,Y]_s \qquad\text{as }n\to\infty,\] for every \(0\le s\le t\).
By this definition, Dirichlet processes form a subclass of the processes admitting quadratic variation. Note, however, that the decomposition \[X=Z+C\] is generally not unique, since any continuous finite variation component may be transferred between \(Z\) and \(C\).
Given a càdlàg process \(X\) and a stopping time \(T\), we define the stopped process \[X^T_t:=X_{t\wedge T}.\] We also define the supremum process \[X_t^*:=\sup_{s\le t}|X_s|.\]
Definition 1. A property of a stochastic process is said to hold locally (respectively pre-locally) if there exists an increasing sequence of stopping times \(T_n\uparrow\infty\) such that the property holds for \(X^{T_n}\) (respectively \(X^{T_n-}\)) for every \(n\).
The following result is taken from [@Collectanea]
Lemma 1. Assume that \(X^1,\ldots,X^n\) admit quadratic variations and that all covariations needed below exist. Then \[\left[\sum_{k=1}^n X^k\right]_t^{1/2} \le \sum_{k=1}^n [X^k]_t^{1/2}.\]
We also have the analogous statement for purely discontinuous quadratic variation.
Lemma 2. Let \(X^1,\ldots,X^n\) be càdlàg processes such that \[\sum_{s\le t}(\Delta X^k_s)^2<\infty, \qquad k=1,\ldots,n.\] Then \[\left( \sum_{s\le t} \left(\sum_{k=1}^n \Delta X^k_s\right)^2 \right)^{1/2} \le \sum_{k=1}^n \left( \sum_{s\le t}(\Delta X^k_s)^2 \right)^{1/2}.\]
Proof. This is triangle inequality in \(\ell^2\). Applied to the sequences \[a^k_s:=\Delta X^k_s, \qquad s\le t,\quad k=1,\ldots,n,\] it gives \[\left( \sum_{s\le t} \left(\sum_{k=1}^n a^k_s\right)^2 \right)^{1/2} \le \sum_{k=1}^n \left( \sum_{s\le t}(a^k_s)^2 \right)^{1/2}.\] Substituting \(a^k_s=\Delta X^k_s\) gives the result. ◻
In this section we introduce the class of Special–Dirichlet processes and establish the basic properties needed in the sequel. In particular, we prove uniqueness of the associated canonical decomposition.
Definition 2. Let \(X\) be a càdlàg adapted process on \([0,t]\). We say that \(X\) is a Special–Dirichlet process if it admits a decomposition \[X = M + \Gamma,\] where \(M\) is a local martingale with \(M_0=0\), and \(\Gamma\) is an adapted càdlàg process such that \[[\Gamma]^c_t = 0,\] and the jumps of \(\Gamma\) are carried by a thin predictable set and satisfy \[\Delta \Gamma_s \in \mathcal{F}_{s-}, \qquad s \le t.\]
The additional predictable jump structure imposed on \(\Gamma\) leads to a canonical decomposition. The remainder of this section is devoted to establishing this fact.
Lemma 3 (Uniqueness of the local martingale component). Let \(X\) be a Special–Dirichlet process on \([0,t]\). Suppose that \[X=M+\Gamma = \widetilde{M}+\widetilde{\Gamma} ,\] where both decompositions satisfy Definition 2, and where \[M_0=\widetilde{M}_0=0 .\] Then \[M=\widetilde{M}\] up to indistinguishability on \([0,t]\).
Proof. Set \[N:=M-\widetilde{M} .\] Then \(N\) is a local martingale and \[N=\widetilde{\Gamma}-\Gamma .\]
Since \[[\Gamma]^c_t=[\widetilde{\Gamma}]^c_t=0,\] we have \[[N]^c_t = [\widetilde{\Gamma}-\Gamma]^c_t = [\widetilde{\Gamma}]^c_t + [\Gamma]^c_t - 2[\widetilde{\Gamma},\Gamma]^c_t.\] Moreover, by the Cauchy–Schwarz inequality for continuous covariation, \[\bigl|[\widetilde{\Gamma},\Gamma]^c_t\bigr| \le [\widetilde{\Gamma}]^c_t{}^{1/2} [\Gamma]^c_t{}^{1/2} = 0.\] Hence \([\widetilde{\Gamma},\Gamma]^c_t=0,\) and therefore \([N]^c_t=0.\)
It remains to show that \(N\) has no jumps. By the definition of a Special–Dirichlet process, the jumps of \(\Gamma\) and \(\widetilde{\Gamma}\) are contained in thin predictable sets, and their jumps are \(\mathcal{F}_{s-}\)-measurable. Hence the jump times of \(\widetilde{\Gamma}-\Gamma\) are carried by a thin predictable set and have \(\mathcal{F}_{s-}\)- measurable jumps.
Let \((T_k)_{k\ge1}\) be predictable stopping times exhausting this thin predictable set. Then, for each \(k\), \[\Delta N_{T_k} = \Delta\widetilde{\Gamma}_{T_k} - \Delta\Gamma_{T_k} \in\mathcal{F}_{T_k-}.\] After localization, the jump is integrable. Since \(N\) is a local martingale and \(T_k\) is predictable, \[\mathbb{E}[\Delta N_{T_k}\mid\mathcal{F}_{T_k-}]=0.\] Because \(\Delta N_{T_k}\) is already \(\mathcal{F}_{T_k-}\)-measurable, we obtain \[\Delta N_{T_k}=0 \qquad\text{a.s.}\] for every \(k\). Thus \(N\) has no jumps.
Consequently, \[[N]_t=[N]^c_t+\sum_{s\le t}(\Delta N_s)^2=0.\] Since \(N_0=0\), it follows that \(N\equiv0\) on \([0,t]\), up to indistinguishability. Hence \(M=\widetilde{M} .\) ◻
The above Lemma motivates the use of the following notation. For a special Dirichlet process \(X\) let \(\mathfrak{M}(X)\) denote its unique local martingale component and let \(\mathfrak{\Gamma}(X)\) denote its unique singular component.
The uniqueness result of the previous section identifies a canonical martingale component \(\mathfrak{M}(X)\) and a canonical singular component \(\mathfrak{\Gamma}(X)\) for every Special–Dirichlet process \(X\). A natural question is whether this decomposition is stable under perturbations of the underlying process. The purpose of this section is to show that convergence in quadratic variation propagates to the canonical decomposition itself.
Lemma 4 (Stability of the Special–Dirichlet decomposition). Let \(X^n\) and \(X\) be Special–Dirichlet processes on \([0,t]\), with canonical decompositions \[X^n=M^n+\Gamma^n, \qquad X=M+\Gamma,\] where \(M^n_0=M_0=0\). Assume that \[[X^n-X]_t \xrightarrow{\mathbb{P}} 0 .\] Then \[[\Gamma^n-\Gamma]_t \xrightarrow{\mathbb{P}} 0\] and \[[M^n-M]_t \xrightarrow{\mathbb{P}} 0 .\]
Proof. Set \[Y^n:=X^n-X,\qquad N^n:=M^n-M,\qquad B^n:=\Gamma^n-\Gamma .\] Then \[Y^n=N^n+B^n .\] By the definition of Special–Dirichlet processes, \(B^n\) has zero continuous quadratic variation and its jump times are carried by a thin predictable set. Moreover, \[\Delta B^n_s\in\mathcal{F}_{s-}\] at every jump time of \(B^n\).
Let \((T^n_k)_{k\ge1}\) be predictable stopping times exhausting the jump times of \(B^n\). Since \[[B^n]^c_t=0,\] we have \[[B^n]_t = \sum_{k\ge1:T^n_k\le t} \bigl(\Delta B^n_{T^n_k}\bigr)^2 .\]
Fix \(\eta>0\), and define the stopping time \[\tau^n_\eta := \inf\{s\le t:[Y^n]_s>\eta\}.\] On the event \(\{\tau^n_\eta>t\}\), we have \([Y^n]_t\le \eta\).
We first estimate the predictable jump part before \(\tau^n_\eta\). After standard localization, we may assume that the local martingale \(N^n\) is square-integrable up to \(t\). For every \(k\), on the event \(\{T^n_k<\tau^n_\eta\}\), the variable \[\Delta B^n_{T^n_k}\] is \(\mathcal{F}_{T^n_k-}\)-measurable. Since \(N^n\) is a local martingale, \[\mathbb{E}\!\left[ \Delta N^n_{T^n_k} \mid \mathcal{F}_{T^n_k-} \right] =0 .\] Hence \[\begin{align} \mathbb{E}\!\left[ \bigl(\Delta Y^n_{T^n_k}\bigr)^2 \mid \mathcal{F}_{T^n_k-} \right] &= \mathbb{E}\!\left[ \bigl(\Delta N^n_{T^n_k} + \Delta B^n_{T^n_k}\bigr)^2 \mid \mathcal{F}_{T^n_k-} \right] \\ &= \bigl(\Delta B^n_{T^n_k}\bigr)^2 + \mathbb{E}\!\left[ \bigl(\Delta N^n_{T^n_k}\bigr)^2 \mid \mathcal{F}_{T^n_k-} \right] \\ &\ge \bigl(\Delta B^n_{T^n_k}\bigr)^2 . \end{align}\] Therefore, \[\mathbb{E}\!\left[ \bigl(\Delta B^n_{T^n_k}\bigr)^2 \mathbf{1}_{\{T^n_k<\tau^n_\eta\}} \right] \le \mathbb{E}\!\left[ \bigl(\Delta Y^n_{T^n_k}\bigr)^2 \mathbf{1}_{\{T^n_k<\tau^n_\eta\}} \right].\] Summing over \(k\) and using monotone convergence gives \[\begin{align} \mathbb{E}\!\left[ \sum_{k\ge1:T^n_k<\tau^n_\eta} \bigl(\Delta B^n_{T^n_k}\bigr)^2 \right] &\le \mathbb{E}\!\left[ \sum_{k\ge1:T^n_k<\tau^n_\eta} \bigl(\Delta Y^n_{T^n_k}\bigr)^2 \right] \\ &\le \mathbb{E}\!\left[ \sum_{s<\tau^n_\eta} \bigl(\Delta Y^n_s\bigr)^2 \right] \\ &\le \eta . \end{align}\] The last inequality follows from the definition of \(\tau^n_\eta\).
By Markov’s inequality, \[\mathbb{P}\!\left( \sum_{k\ge1:T^n_k<\tau^n_\eta} \bigl(\Delta B^n_{T^n_k}\bigr)^2 >\varepsilon \right) \le \frac{\eta}{\varepsilon}.\] Hence \[\begin{align} \mathbb{P}\!\left([B^n]_t>\varepsilon\right) &\le \mathbb{P}(\tau^n_\eta\le t) + \mathbb{P}\!\left( \sum_{k\ge1:T^n_k<\tau^n_\eta} \bigl(\Delta B^n_{T^n_k}\bigr)^2 >\varepsilon \right) \\ &\le \mathbb{P}([Y^n]_t>\eta) + \frac{\eta}{\varepsilon}. \end{align}\] Since \([Y^n]_t=[X^n-X]_t\to0\) in probability, we obtain \[\limsup_{n\to\infty} \mathbb{P}\!\left([B^n]_t>\varepsilon\right) \le \frac{\eta}{\varepsilon}.\] Letting \(\eta\downarrow0\), we conclude that \[[B^n]_t=[\Gamma^n-\Gamma]_t \xrightarrow{\mathbb{P}}0 .\]
It remains to prove the convergence of the martingale parts. Since \[N^n=Y^n-B^n,\] we have, by the triangle inequality for quadratic variation, \[[N^n]_t^{1/2} \le [Y^n]_t^{1/2} + [B^n]_t^{1/2}.\] Therefore \[[M^n-M]_t=[N^n]_t \xrightarrow{\mathbb{P}}0 .\] ◻
Having established stability of the canonical decomposition, we now turn to transformation theory. The main question is whether the Special–Dirichlet structure is preserved under nonsmooth changes of variables. Our first result shows that the class is closed under a broad family of transformations and provides an explicit decomposition of the transformed process.
For a function \(f:\mathbb{R}\to\mathbb{R}\) we define \[f'(x) = \limsup_{h\downarrow0} \frac{f(x+h)-f(x)}{h}.\] Throughout the paper, \(f'\) refers to this upper Dini derivative.
Theorem 3 (Closure under nonsmooth transformations). Let \(X\) be a Special–Dirichlet process on \([0,t]\), with canonical decomposition \[X=M+\Gamma, \qquad M_0=0,\] where \(M\) is a local martingale and \(\Gamma\) is an adapted càdlàg process such that \([\Gamma]^c_t=0,\) and whose jumps are carried by a thin predictable set and satisfy \(\Delta\Gamma_s\in\mathcal{F}_{s-}.\) Let \(f\) be the primitive of a locally bounded function \(f'\) with an at most countable set of discontinuities. Assume that \[\int_0^t \mathbf{1}_{\{X_s\notin\operatorname{diff}(f)\}}\,d[X]^c_s=0.\] Then \(f(X)\) is again a Special–Dirichlet process.
More precisely, \[f(X)=M^f+\Gamma^f,\] where \[M^f_t := \int_0^t f'(X_{s-}+\Delta\Gamma_s)\,dM^c_s + \int_0^t\int_{\mathbb{R}} \left(f(X_{s-}+\Delta\Gamma_s+x)-f(X_{s-}+\Delta\Gamma_s)\right)(\mu^M-\nu^M)(ds,dx)\] is a local martingale, and \(\Gamma^f:=f(X)-M^f\) satisfies \([\Gamma^f]^c_t=0.\)
Moreover, \[\Delta\Gamma^f_s = f(X_{s-}+\Delta\Gamma_s)-f(X_{s-}) + \int_{\mathbb{R}} \left(f(X_{s-}+\Delta\Gamma_s+x)-f(X_{s-}+\Delta\Gamma_s)\right)\nu^M(\{s\},dx).\] In particular, the jumps of \(\Gamma^f\) are carried by a thin predictable set and satisfy \[\Delta\Gamma^f_s\in\mathcal{F}_{s-}.\]
Proof. Write \[\widehat X_s:=X_{s-}+\Delta\Gamma_s.\] Since \(X_{-}\) is predictable and the jump times of \(\Gamma\) are carried by a thin predictable set with \(\mathcal{F}_{s-}\)-measurable jumps, \(\widehat X\) is predictable. Let \[\Phi(s,x):=f(\widehat X_s+x)-f(\widehat X_s).\] After localization, we may assume that \(X\), \(M\), and \(\Gamma\) are bounded on \([0,t]\), and that \(f'\) is bounded on the relevant compact interval. Therefore, on the relevant compact interval, there exists \(C>0\) such that, whenever both \[\widehat X_s \quad\text{and}\quad \widehat X_s+x\] belong to this interval, \[|\Phi(s,x)| \le C|x|.\] Since \(M^c\) is a continuous local martingale, \[\int_0^\cdot f'(\widehat X_s)\,dM^c_s\] is a local martingale, and \[\int_0^\cdot\int_{\mathbb{R}} \Phi(s,x)(\mu^M-\nu^M)(ds,dx)\] is a well-defined purely discontinuous local martingale. Hence \(M^f\) is a local martingale.
Set \[\Gamma^f:=f(X)-M^f.\]
We first compute the jumps of \(\Gamma^f\). Put \[u_s:=\Delta M_s, \qquad v_s:=\Delta\Gamma_s, \qquad y_s:=X_{s-}.\] Then \[\widehat X_s=y_s+v_s, \qquad X_s=\widehat X_s+u_s.\] Thus \[\Delta f(X)_s = f(\widehat X_s+u_s)-f(y_s).\] On the other hand, since the first integral in \(M^f\) is continuous, \[\begin{align} \Delta M^f_s &= \Phi(s,u_s) - \int_{\mathbb{R}}\Phi(s,x)\nu^M(\{s\},dx) \\ &= f(\widehat X_s+u_s)-f(\widehat X_s) - \int_{\mathbb{R}}\Phi(s,x)\nu^M(\{s\},dx). \end{align}\] Therefore \[\begin{align} \Delta\Gamma^f_s &= \Delta f(X)_s-\Delta M^f_s \\ &= f(\widehat X_s)-f(y_s) + \int_{\mathbb{R}}\Phi(s,x)\nu^M(\{s\},dx) \\ &= f(X_{s-}+\Delta\Gamma_s)-f(X_{s-}) + \int_{\mathbb{R}} \Phi(s,x)\nu^M(\{s\},dx). \end{align}\] The first term is \(\mathcal{F}_{s-}\)-measurable by the defining jump condition on \(\Gamma\). The second term is \(\mathcal{F}_{s-}\)-measurable because \(\nu^M\) is predictable and the integrand is predictable. Hence \[\Delta\Gamma^f_s\in\mathcal{F}_{s-}.\] The jumps of \(\Gamma^f\) are supported on the union of the jump times of \(\Gamma\) and the predictable atoms of \(\nu^M\), hence on a thin predictable set.
It remains to prove that \[[\Gamma^f]^c_t=0.\]
Since \(f'\) is a locally bounded Baire class \(1\) function, there exist polynomials \(p_n\) such that \[p_n(x)\to f'(x)\] for every \(x\in[-m,m]\). Moreover, since \(f'\) is bounded on \([-m,m]\), the approximation may be chosen so that \[\sup_n\sup_{|x|\le m}|p_n(x)|<\infty.\] Define \[f_n(x):=f(-m)+\int_{-m}^x p_n(u)\,du,\] and \[\Phi_n(s,x):=f_n(\widehat X_s+x)-f_n(\widehat X_s).\]
Let \(M^n\) be defined as \(M^f\) with \(f',\Phi\) replaced by \(p_n,\Phi_n\), namely \[M^n_t := \int_0^t p_n(\widehat X_s)\,dM^c_s + \int_0^t\int_{\mathbb{R}} \Phi_n(s,x)(\mu^M-\nu^M)(ds,dx),\] and set \[\Gamma^n:=f_n(X)-M^n.\]
Step 1: smooth case. Since \(f_n\in C^1\), the standard change-of-variables formula applied to \(X=M+\Gamma\), together with \([\Gamma]^c=0\), yields a decomposition in which the residual satisfies \([\Gamma^n]^c_t=0.\)
Step 2: decomposition of the difference. Write \[\Gamma^f = \bigl(f(X)-f_n(X)\bigr) - (M^f-M^n) + \Gamma^n.\]
By the triangle inequality for continuous quadratic variation, \[\begin{align} [\Gamma^f]_t^{c\,1/2} &\le [f(X)-f_n(X)]_t^{c\,1/2} + [M^f-M^n]_t^{c\,1/2} + [\Gamma^n]_t^{c\,1/2}. \end{align}\] Since \([\Gamma^n]^c_t=0\), it suffices to show that the first two terms converge to zero.
Step 3: control of \(f(X)-f_n(X)\). Set \(h_n:=f-f_n\). Then \(h_n\) is the primitive of \(f'-p_n\). By Lemma 6.2 in [@Collectanea] and the non-charging assumption, \[[h_n(X)]^c_t = \int_0^t \bigl(f'(X_s)-p_n(X_s)\bigr)^2\,d[X]^c_s.\]
After localization, assume \(|X_s|\le m\) and \[|f'(x)|\le K, \qquad |p_n(x)|\le K \quad (|x|\le m).\] Then \[|f'(X_s)-p_n(X_s)|^2 \le 4K^2,\] and since \(p_n(x)\to f'(x)\) for every \(x\in[-m,m]\), we have, for every \(\omega\) and every \(s\le t\) on the localized set, \[(f'(X_s(\omega))-p_n(X_s(\omega)))^2\to0.\] Since \(d[X]^c_s\) is a finite measure on \([0,t]\), dominated convergence gives \[[f(X)-f_n(X)]^c_t=[h_n(X)]^c_t\longrightarrow0.\]
Step 4: control of the martingale difference. We have \[M^f_t-M^n_t = \int_0^t (f'(\widehat X_s)-p_n(\widehat X_s))\,dM^c_s + \int_0^t\int_{\mathbb{R}} (\Phi-\Phi_n)(s,x)(\mu^M-\nu^M)(ds,dx).\] The second term is purely discontinuous, hence \[[M^f-M^n]^c_t = \int_0^t (f'(\widehat X_s)-p_n(\widehat X_s))^2\,d[M]^c_s.\]
Since \([M]^c\) is continuous, the measure \(d[M]^c_s\) has no atoms. The jump set of the càdlàg process \(\Gamma\) is at most countable on \([0,t]\). Hence \(d[M]^c_s\) does not charge the set \(\{s:\Delta\Gamma_s\ne0\}\). Consequently, \[\widehat X_s=X_{s-}+\Delta\Gamma_s=X_{s-} \quad d[M]^c_s\text{-a.e.}\] Moreover, \(d[M]^c_s=d[X]^c_s\), since \([\Gamma]^c=0\) and the continuous quadratic variation of \(X\) is carried by the continuous martingale part of \(M\). Therefore the same dominated convergence argument gives \[[M^f-M^n]^c_t \longrightarrow 0.\]
Step 5: conclusion. Combining the estimates, \([\Gamma^f]_t^{c\,1/2}\to0,\) hence \([\Gamma^f]^c_t=0.\) ◻
The preceding theorem applies to general Special–Dirichlet processes. For special semimartingales, Lowther’s non-charging condition is automatically satisfied, yielding the following immediate consequence.
Corollary 1. Let \(Z\) be a special semimartingale on \([0,t]\), and let \(f \in C^1(\mathbb{R})\). Then \(f(Z)\) is a special Dirichlet process.
Proof. This follows from the above theorem and the fact that \[\int_0^t \mathbf{1}_{\{X_s\notin\operatorname{diff}(f)\}}\,d[X]^c_s=0\] is true whenever \(X\) is a semimartingale (as was shown in [@Low]). ◻
We now combine the structural stability result of Section 4 with the transformation theorem above. The goal is to show that convergence of the underlying processes propagates through the transformation and remains visible at the level of the canonical decomposition. As mentioned earlier, for a special Dirichlet process X let \(\mathfrak{M}(X)\) denote its unique local martingale component and let \(\mathfrak{\Gamma}(X)\) denote its unique predictable component. We shall need two lemmas to prove the theorem in this section. The first lemma shows that convergence in the \(J_1\)-topology, combined with vanishing quadratic variation of the error process, implies convergence of the corresponding left limits.
Lemma 5. Let \(X^n\) and \(X\) be real-valued cadlag functions on \([0,t]\). Assume that \[X^n \to X \quad\text{in the } J_1\text{-topology}\] and that \[[X^n-X]_t \to 0 .\] Then, for every \(s\in(0,t]\), \[X^n_{s-}\to X_{s-}.\]
Proof. Fix \(s\in(0,t]\), and write \[L:=X_{s-},\qquad R:=X_s .\] We use the following standard consequence of \(J_1\)-convergence: if \(x_n\to x\) in the \(J_1\)-topology and \(u_n\to s\), then every cluster point of \(x_n(u_n)\) belongs to the completed graph segment \[[[x(s-),x(s)]] := \{\theta x(s-)+(1-\theta)x(s):\theta\in[0,1]\}.\]
Put \[Y^n:=X^n-X .\] Since the quadratic variation contains the squared jumps, \[\bigl(\Delta Y^n_s\bigr)^2\le [Y^n]_t .\] Hence \(\Delta Y^n_s\to0 .\) Equivalently, \[\Delta X^n_s-\Delta X_s\to0,\] and therefore \[X^n_s-X^n_{s-}\to R-L .\]
Let \(a\) be an arbitrary cluster point of \(X^n_{s-}\). Passing to a subsequence, we may assume that \(X^n_{s-}\to a .\) For each \(n\), choose \[u_n\in (s-1/n,s)\cap[0,t]\] such that \[|X^n(u_n)-X^n_{s-}|<1/n .\] This is possible by the existence of the left limit \(X^n_{s-}\). Then \(u_n\to s\), and along the chosen subsequence \[X^n(u_n)\to a .\] By the completed graph consequence of \(J_1\)-convergence, \(a\in [[L,R]] .\)
Passing to a further subsequence if necessary, assume also that \(X^n_s\to b .\) Again by the completed graph consequence, now with \(u_n=s\), we have \(b\in [[L,R]] .\) On the other hand, \[b-a = \lim_{n\to\infty} (X^n_s-X^n_{s-}) = R-L .\] Since \(a,b\in [[L,R]]\) and \(b-a=R-L\), it follows in the real-valued case that \[a=L,\qquad b=R .\] Thus every cluster point of \(X^n_{s-}\) equals \(X_{s-}\). Consequently, \(X^n_{s-}\to X_{s-}.\) ◻
The second lemma, taken from [@CompJump], provides a threshold isolation principle: for arbitrarily small jump-size thresholds, one can find a neighborhood around the threshold which, with probability tending to one, eventually contains no jumps of either \(X\) or \(X^n\).
Lemma 6 (Threshold Isolation Principle). Let \(X\) and \(\{X^n\}_{n\ge1}\) be càdlàg processes on \([0,t]\) such that \[[X^n-X]_t \to 0 \qquad \text{in probability}.\] Fix \(r>0\), and for each \(k\ge1\) define \[\delta(k,r):=\frac{r}{2^{k+2}}, \qquad L(r,k):=r\Bigl(1-\frac{3}{2^{k+2}}\Bigr),\] and \[A_k(r) := \Bigl\{ \omega: \bigl||\Delta X_s(\omega)|-L(r,k)\bigr|>\delta(k,r) \text{ for all } s\le t \Bigr\}.\] Then \[\mathbb{P}\Bigl(\bigcup_{k\ge1}A_k(r)\Bigr)=1.\]
Moreover, for each fixed \(k\ge1\), if we define \[B_n(k,r) := \Bigl\{ \sup_{s\le t}|\Delta(X^n-X)_s|<\delta(k,r) \Bigr\},\] then \(\lim_{n\to\infty}\mathbb{P}\bigl(B_n(k,r)\bigr)=1.\) On the event \(A_k(r)\cap B_n(k,r)\), the classification of jumps relative to the threshold \(L(r,k)\) is preserved, i.e. \[|\Delta X_s|\le L(r,k)-\delta(k,r) \quad\Longrightarrow\quad |\Delta X^n_s|<L(r,k),\] and \[|\Delta X_s|\ge L(r,k)+\delta(k,r) \quad\Longrightarrow\quad |\Delta X^n_s|>L(r,k),\] for all \(s\le t\).
In particular, on \(A_k(r)\cap B_n(k,r)\), \[1_{\{|\Delta X^n_s|\le L(r,k)\}} \le 1_{\{|\Delta X_s|\le L(r,k)\}} \qquad \text{for all } s\le t,\] and therefore \[\sum_{s\le t} (\Delta X^n_s)^2 1_{\{|\Delta X^n_s|\le L(r,k)\}} \le 2[X^n-X]_t + 2\sum_{s\le t} (\Delta X_s)^2 1_{\{|\Delta X_s|\le L(r,k)\}}.\]
With these tools we can now prove the following.
Theorem 4. Let \(f\in C^1\). Let \(\{X^n\}_n\) be special Dirichlet processes such that for each \(n\), \(X^n\) and \(X\) have quadratic variations along the same refining sequence, that \(X^n\xrightarrow{J_1}X\) in probability, \([X^n-X]_t\xrightarrow{{\mathbb{P}}}0\). Then
\([\mathfrak{\Gamma}(f(X^n))-\mathfrak{\Gamma}(f(X))]_t \xrightarrow{{\mathbb{P}}} 0\) and \(\mathfrak{M}(f(X^n))\xrightarrow{ucp}\mathrm{M}(f(X))\) as \(n\to
\infty\).
Proof. We first establish that \([f(X^n)-f(X)]_t\xrightarrow{{\mathbb{P}}}0\). According to Lemma 4 it then follows that \([\mathfrak{M}(f(X^n))-\mathfrak{M}(f(X))]_t\xrightarrow{{\mathbb{P}}}0\) and \([\mathfrak{\Gamma}(f(X^n))-\mathfrak{\Gamma}(f(X))]_t\xrightarrow{{\mathbb{P}}}0\). By localization and the Burkholder-David-Gundy inequality it then follows that \(\mathfrak{M}(f(X^n))\xrightarrow{ucp}\mathfrak{M}(f(X))\).
Let \(\{n_j\}_{j\in{\mathbb{N}}}\) be an arbitrary subsequence and extract a further subsequence \(\{n_{j_l}\}_{l\in{\mathbb{N}}}\) such that both \([X^{n_{j_l}}-X]_t\to 0\) a.s. and \(X^{n_{j_l}}\xrightarrow{J_1}X\) a.s.. according to Lemma 5 we conclude that \[X^{n_{j_l}}_{s-}\xrightarrow{a.s.}X_{s-} \qquad\text{for every } s\in[0,t].\] Since the sequence \(\{n_j\}_{j\in{\mathbb{N}}}\) is arbitrary, it suffices to show that \([f(X^{n_{j_l}})-f(X)]_t\xrightarrow{{\mathbb{P}}}0\) as \(l\to \infty\), so by relabelin we may assume that \(X^{n}_{s-}\xrightarrow{a.s.}X_{s-}\). If we let \[B_R=\left\{\sup_n(X^n)^*_t\vee X^*_t\le R\right\}\] then \[{\mathbb{P}}\left(\cup_{R} B_R\right)=\lim_{R\to\infty}{\mathbb{P}}(B_R)=1,\] so we may assume that \(X,\Delta X\), \(\{X^n\}_n\) and \(\{\Delta X^n\}_n\) are all uniformly bounded by the constant \(R\). We will let \(L=\sup_{x\in[-R-a,R+a]} |f'(x)|\) (which must be finite as cadlag functions are bounded on compacts). According to Theorem 2.1 in [@Low] \[\begin{align} f(X_s)=\int_0^sf'(X_{u-})dM_u+V_s \end{align}\] and similarly \[\begin{align} f(X^n_s)=\int_0^sf'(X^n_{u-})dM^n_u+V^n_s \end{align}\]
Let \(X^n=M^n+\Gamma^n\) and \(X=M+\Gamma\) be the canonical decompositions of \(X\) and \(X^n\) respectively. Using Lemma 2.5 in [@Collectanea], \[\begin{align} \label{fuljavel} [f(X^n)-f(X)]_t^{\frac{1}{2}} &\le \left[\int_0^. f'(X_{s-})dM_s-\int_0^. f'(X^n_{s-})dM^n_s\right]_t^{\frac{1}{2}} + \left[V^n-V\right]_t^{\frac{1}{2}} \end{align}\tag{3}\] for the first term on the right-hand side of 3 we have, due to Lemma 1 \[\begin{align} \label{Mterms} \left[\int_0^. f'(X_{s-})dM_s-\int_0^. f'(X^n_{s-})dM^n_s\right]_t^{\frac{1}{2}} &\le \left[\int_0^. \left(f'(X_{s-})- f'(X^n_{s-})\right) dM_s\right]_t^{\frac{1}{2}} + \left[\int_0^. \left(f'(X^n_{s-})\right) d(M_s-M^n_s)\right]_t^{\frac{1}{2}}\nonumber \\ &= \left( \int_0^t \left(f'(X_{s-})- f'(X^n_{s-})\right)^2 d[M]_s\right)^{\frac{1}{2}} + \left(\int_0^t f'(X^n_{s-})^2 d[M-M^n]_s\right)_t^{\frac{1}{2}}. \end{align}\tag{4}\] On the set \(B_R\) we have that \(\left(f'(X_{s-})- f'(X^n_{s-})\right)^2\le 4L^2\) so it follows by dominated convergence (applied pathwise) that \[\lim_{n\to\infty}\left( \int_0^t \left(f'(X_{s-})- f'(X^n_{s-})\right)^2 d[M]_s\right)^{\frac{1}{2}}1_{B_R}=0.\] Hence, for every \(\epsilon>0\) \[{\mathbb{P}}\left( \left( \int_0^t \left(f'(X_{s-})- f'(X^n_{s-})\right)^2 d[M]_s\right)^{\frac{1}{2}}\ge \epsilon\right) \le {\mathbb{P}}\left(\left( \int_0^t \left(f'(X_{s-})- f'(X^n_{s-})\right)^2 d[M]_s\right)^{\frac{1}{2}}1_{B_R}\ge \epsilon\right) + {\mathbb{P}}(B_R^c)\] where the first term converges to zero as \(n\to\infty\) and the second term as \(R\to \infty\). For the second term on the right-hand side of 4 , note that \([M-M^n]_t\xrightarrow{{\mathbb{P}}}0\) by Lemma 4 and therefore \[{\mathbb{P}}\left( \left(\int_0^t f'(X^n_{s-})^2 d[M-M^n]_s\right)_t^{\frac{1}{2}}\ge \epsilon\right) \le {\mathbb{P}}\left( L[M-M^n]_t^{\frac{1}{2}}\ge \epsilon\right) + {\mathbb{P}}(B_R^c)\] which converges to zero. This takes care of the first term on the right-hand side of 3 . For the second term on the right-hand side of note that, by Lemma 2 \[\begin{align} [V^n-V]^{\frac{1}{2}}_t=\left([V^n-V]^d_t\right)^{\frac{1}{2}} \le \left[\int_0^. f'(X_{s-})dM_s-\int_0^. f'(X^n_{s-})dM^n_s\right]_t^{\frac{1}{2}} + \left([f(X^n)-f(X)]^d_t\right)^{\frac{1}{2}}. \end{align}\] We have already shown that the first term on the right-hand side above converges to zero so to finish the proof it only remains to prove that also the second term vanishes. Set \[F^n:=f(X^n)-f(X), \qquad D^n:=X^n-X .\] We work on the localized set \(B_R\). On this set \(f\) is Lipschitz on the relevant compact interval; let \(K_R\) denote a corresponding Lipschitz constant. Thus, whenever both arguments stay in the localized range, \[|f(x)-f(y)|\le K_R |x-y|.\]
Fix \(r>0\). By Lemma 6, choose \(k\ge1\) such that the event \(A_k(r)\) has probability arbitrarily close to one, and put \[\rho:=L(r,k).\] On \(A_k(r)\cap B_n(k,r)\), the classification of jumps above and below the threshold \(\rho\) is preserved. In particular, \[|\Delta X^n_s|>\rho \quad\Longleftrightarrow\quad |\Delta X_s|>\rho .\] Let \[J_\rho:=\{s\le t:|\Delta X_s|>\rho\}.\] Since \(X\) has finite quadratic variation, \(J_\rho\) is finite.
We split \[[F^n]^d_t = \sum_{s\in J_\rho}(\Delta F^n_s)^2 + \sum_{s\le t:\,|\Delta X_s|\le \rho}(\Delta F^n_s)^2\] on the event \(A_k(r)\cap B_n(k,r)\).
First consider the large-jump part. If \(s\in J_\rho\), then by Lemma 5, \[X^n_{s-}\to X_{s-}.\] Moreover, \[\Delta X^n_s-\Delta X_s = \Delta D^n_s\to0,\] because \[|\Delta D^n_s|^2\le [D^n]_t\to0.\] Hence also \(X^n_s\to X_s\). Since \(f\) is continuous, \[\Delta F^n_s = \Delta f(X^n)_s-\Delta f(X)_s \longrightarrow 0\] for every \(s\in J_\rho\). Since \(J_\rho\) is finite, \[\sum_{s\in J_\rho}(\Delta F^n_s)^2\longrightarrow 0.\]
It remains to control the small-jump part. On \(A_k(r)\cap B_n(k,r)\), if \(|\Delta X_s|\le \rho\), then \(|\Delta X^n_s|\le \rho\). Therefore, using the Lipschitz bound, \[\begin{align} |\Delta F^n_s| &= \left| \bigl(f(X^n_s)-f(X^n_{s-})\bigr) - \bigl(f(X_s)-f(X_{s-})\bigr) \right| \\ &\le K_R |\Delta X^n_s| + K_R |\Delta X_s|. \end{align}\] Consequently, \[(\Delta F^n_s)^2 \le 2K_R^2\bigl((\Delta X^n_s)^2+(\Delta X_s)^2\bigr).\] Summing over the small jumps gives \[\begin{align} \sum_{s\le t:\,|\Delta X_s|\le \rho}(\Delta F^n_s)^2 &\le 2K_R^2 \sum_{s\le t:\,|\Delta X_s|\le \rho}(\Delta X^n_s)^2 + 2K_R^2 \sum_{s\le t:\,|\Delta X_s|\le \rho}(\Delta X_s)^2. \end{align}\] Since \[\Delta X^n_s=\Delta X_s+\Delta D^n_s,\] we have \[(\Delta X^n_s)^2 \le 2(\Delta X_s)^2+2(\Delta D^n_s)^2.\] Thus \[\begin{align} \sum_{s\le t:\,|\Delta X_s|\le \rho}(\Delta F^n_s)^2 &\le 6K_R^2 \sum_{s\le t:\,|\Delta X_s|\le \rho}(\Delta X_s)^2 + 4K_R^2 \sum_{s\le t}(\Delta D^n_s)^2 \\ &\le 6K_R^2 \sum_{s\le t:\,|\Delta X_s|\le \rho}(\Delta X_s)^2 + 4K_R^2 [D^n]_t . \end{align}\] Letting \(n\to\infty\), the second term vanishes. Hence, on \(A_k(r)\cap B_n(k,r)\), \[\limsup_{n\to\infty}[F^n]^d_t \le 6K_R^2 \sum_{s\le t:\,|\Delta X_s|\le \rho}(\Delta X_s)^2.\] Finally, as \(\rho\downarrow0\), \[\sum_{s\le t:\,|\Delta X_s|\le \rho}(\Delta X_s)^2 \longrightarrow0,\] because \(\sum_{s\le t}(\Delta X_s)^2<\infty.\)
Since \(r>0\) was arbitrary and the events \(A_k(r)\cap B_n(k,r)\) have probability tending to one, it follows that \[[f(X^n)-f(X)]^d_t = [F^n]^d_t \xrightarrow{\mathbb{P}}0.\]
Combining this with the previous estimates yields \[[f(X^n)-f(X)]_t\xrightarrow{\mathbb{P}}0.\] This completes the proof. ◻
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Philip Kennerberg Independent researcher, Sweden Email: pkennerberg@gmail.com↩︎