October 31, 2025
We study a singular stochastic equation driven by regular noise of fractional Brownian type with Hurst index \(H \in (1,\infty)\setminus\mathbb{Z}\) and drift coefficient \(b \in \mathcal{C}^\alpha\), where \(\alpha > 1 - \tfrac{1}{2H}\). The strong well-posedness of this equation was first established in [1], a phenomenon known as regularization by regular noise. In this note, we provide a numerical analysis of the equation. Specifically, we prove that the Euler–Maruyama approximation \(X^n\) converges strongly to the unique solution \(X\) at rate \(n^{-1}\). Moreover, we show that \(n(X - X^n)\) converges in probability to a non-trivial limit as \(n \to \infty\), which confirms that the rate \(n^{-1}\) is optimal for this scheme. In this sense, this provides a first-order numerical method for equations with non-Lipschitz drift while still achieving the rate \(n^{-1}\).
It is known that when \(H\in(0,1)\), a \(d\)-dimensional fractional Brownian motion (fBM) \(B^H\) can be defined via the Mandelbrot–van Ness representation [2]: \[\begin{align} \label{def:fBM-Hsmall1} B_t^H:=\int_{-\infty}^0(|t-s|^{H-\frac{1}{2}}-|s|^{H-\frac{1}{2}})\mathrm{d} W_s+\int_0^t|t-s|^{H-\frac{1}{2}}\mathrm{d} W_s, \quad t\geqslant 0, \end{align}\tag{1}\] where \(W\) is a two-sided \(d\)-dimensional standard Brownian motion on some probability space \((\Omega,{\mathcal{F}},{\mathbb{P}})\). As discussed in [1], the fractional integral in 1 can in fact be naturally extended to the regime where the Hurst parameter \(H>1\), that is, for \(H\in (1,\infty)\backslash{\mathbb{Z}}\), \[\begin{align} \label{def:fbmHbig1} B_t^H:=\int_{0\leqslant s_1\leqslant\ldots\leqslant s_{\lfloor H\rfloor}\leqslant t}B^{H-\lfloor H\rfloor}_{s_1}\mathrm{d} {s_1}\ldots \mathrm{d} s_{\lfloor H\rfloor}. \end{align}\tag{2}\] Due to the possible multiple integrals in 2 , we can see clearly that the paths of \(B^H\) for \(H>1\) are regular, at least \(C^1\). This fact leads to one of the appealing results of [1], which shows the strong well-posedness of the following equation with singular \(b\): \[\begin{align} \label{eq:SDE} \mathrm{d} X_t=b(X_t)\mathrm{d} t+\mathrm{d} B_t^H,\quad X_0=x_0\in{\mathbb{R}}^d, \end{align}\tag{3}\] where \(b\in C^\alpha({\mathbb{R}}^d;{\mathbb{R}}^d)\) with \(\alpha\in(1-\frac{1}{2H},1)\) and \(B^H\) is a \(d\)-dimensional fBM with \(H\in(1,\infty)\backslash{\mathbb{Z}}\). This result supports the principle of regularization by noise: increased noise roughness leads to enhanced regularization([3]–[9]). There is also an alternative way, mentioned in [1], to interpret 3 as a singular coupled equation perturbed by degenerate noise: \[\begin{align} \label{eq:SDEcoup} \left\{ \begin{array}{cc} \mathrm{d} X_t & =\big(b(X_t)+V_t^{\lfloor H\rfloor}\big) \mathrm{d} t,\\ \mathrm{d} V_t^{\lfloor H\rfloor}&=V_t^{\lfloor H\rfloor-1}\mathrm{d} t,\\ \qquad\qquad\qquad\qquad \ldots, &\\ \mathrm{d} V_t^1&=\mathrm{d} B_t^{H-\lfloor H\rfloor}, \end{array}\right. \end{align}\tag{4}\] which shares a similar framework with [10]–[13], but with non-Markovian noise when \(H\neq k+\frac{1}{2}\) for any \(k\in{\mathbb{N}}\).
Our motivation here is to provide a numerical result for this equation by considering its Euler–Maruyama (EM) scheme \[\begin{align} \label{eq:SDE-EM} \mathrm{d} X_t^n=b(X_{k_n(t)}^n)\mathrm{d} t+\mathrm{d} B_t^H,\qquad X_0^n=x_0^n\in{\mathbb{R}}^d \end{align}\tag{5}\] with \(k_n(t)\mathrel{\vcenter{:}}=\frac{\lfloor nt\rfloor}{n}\).
When \(H\in (0,1)\), [14] has shown the strong convergence rate \(n^{-(\frac{1}{2}+\alpha H)\wedge 1+\epsilon}\) of the scheme 5 to 3 , in which the Girsanov Theorem and the Stochastic Sewing Lemma (SSL) [6] play crucial roles. More precisely, the methodology in [14] can be roughly summarized as follows (taking \(x_0=x_0^n\)): \[\begin{align} \|\sup_{t\in[0,1]}|X_t-X^n_t|\|_{L^p_\omega}\overset{\text{Girsanov}}{\lesssim}\big\|\int_0^1b(B_{s}^H)-b(B_{k_n(s)}^H)\mathrm{d} r\big\|_{L^p_\omega}\overset{\text{SSL}}{\lesssim} n^{-(\frac{1}{2}+\alpha H)\wedge 1+\epsilon}. \end{align}\]
When \(H>1\), on the one hand, as indicated in [1], [9], the Girsanov Theorem becomes less helpful; meanwhile, PDE tools clearly do not apply. Therefore, the challenge in showing the convergence of 5 to 3 lies in two aspects compared with known results: the absence of Girsanov’s theorem and the non-Markovian nature of the noise. Alternatively, although [1] studies only well-posedness, it already hints at a possible way to avoid using Girsanov’s theorem; concerning numerical approximation, [15] provides similar evidence, but it considers singular SDEs driven by an \(\alpha\)-stable process, which is Markovian. Nevertheless, we are able to show a convergence rate of \(n^{-1}\), which is comparable with [14], since \(H>1\) here implies \(\alpha>1-\frac{1}{2H}>\frac{1}{2}\).
Given this convergence rate, it is natural to ask how far it is from being optimal. Such questions on the optimality of the EM scheme have been addressed, for instance, in [16]–[18] for Brownian noise, [19] for Lévy processes with jumps, and [20]–[23] for fBM with \(H<1\). In particular, [20] confirms that for \(H\in(\frac{1}{2},1)\), the order \(n^H\) is optimal for the EM scheme 5 when \(b\in {\mathcal{C}}^2\) (twice differentiable) with possible linear growth. Here, we complete this result with rate \(n^{-1}\) for \(H>1\) and less regular \(b\) (in fact, only \({\mathcal{C}}^1\)). The idea of the proof is straightforward: we show that for \(b\in{\mathcal{C}}^\alpha\), the following approximation converges in probability to a possibly non-zero limit: \[\begin{align} n(X_t-X^n_t)\overset{n\rightarrow\infty}{\rightarrow} c(t)\neq0, \end{align}\] which indicates that for 5 , the best convergence rate one can expect is no faster than order \(n^{-1}\). Consequently, this also verifies that the rate we obtain is indeed optimal. Evidently, 5 provides a stochastic numerical method for simulating equations with singular \(b\) while still converging with rate \(n^{-1}\).
Lastly, we mention a few related works that share a similar interest and spirit in the study of numerical approximations. For equations of the type 3 with singular drift, convergence results have been established in [24]–[27] for additive Brownian motion, in [14], [28]–[30] for multiplicative Brownian noise, in [15], [31] for Lévy processes with jumps, and in [14], [32] for fractional Brownian motion. A slightly different notion of singularity-namely, piecewise Lipschitz coefficients-has also been investigated, with convergence results obtained in works such as [19], [33]–[35]. We emphasize that this is only a brief selection from a vast body of literature, and we encourage readers to consult the cited works for further details and insights.
In 2 we introduce the necessary notations and main results. 3 collects all of the crucial tools and properties of fBM. We present the central analysis and the proof of the main convergence result in 4. Finally, we show optimality in 5. 6 contains several auxiliary proofs.
On finite dimensional vector spaces we always use the Euclidean norm.
For \(k\in\mathbb{N}\), \(f:\mathbb{R}^d\mapsto \mathbb{R}\), denote \(\partial_k f(x)\mathrel{\vcenter{:}}=\frac{\partial f(x)}{\partial x_k}\) for \(x\in\mathbb{R}^d\) and \(\nabla f(x)\mathrel{\vcenter{:}}=(\partial_if(x))_{1\leqslant i\leqslant d}\), the derivative is understood in the weak sense. For vector-valued \(f\) we use the same notation, and \(\nabla^k f\) is defined via \(\nabla(\nabla ^{k-1}f)\) iteratively. For a multi-index \(k=(k_1,\ldots,k_d)\in{\mathbb{N}}^d\), denote \(\partial^k f(x)\mathrel{\vcenter{:}}=\frac{\partial^{|k|} f(x)}{\partial x_{k_1}\cdots\partial x_{k_d}}\). If \(k=(0,\ldots,0)\), we use convention \(\partial^kf=f\). We denote by \({{\mathcal{C}}}^\infty_0\) (\({{\mathcal{C}}}_p^\infty\), resp.) the set of all continuously infinitely differentiable functions that, along with all of their partial derivatives, are compactly supported (of polynomial growth, resp.).
For \(\alpha\in(0,1)\), we set \(\mathcal{C}^\alpha({\mathbb{R}}^d)\) to be the space of continuous functions such that \[\begin{align} \Vert f\Vert_{\mathcal{C}^\alpha}\mathrel{\vcenter{:}}=[f]_{\mathcal{C}^\alpha}+\sup_{x\in{\mathbb{R}}^d}|f(x)|\mathrel{\vcenter{:}}=\sup_{x,y\in{\mathbb{R}}^d,x\neq y}\frac{|f(x)-f(y)|}{|x-y|^\alpha}+\sup_{x\in{\mathbb{R}}^d}|f(x)|<\infty. \end{align}\] Here, and often below, we write \(\mathcal{C}^\alpha\) instead of \(\mathcal{C}^\alpha({\mathbb{R}}^d)\) for simplicity. For \(\alpha\in(0,\infty)\), we define \(\mathcal{C}^\alpha({\mathbb{R}}^d)\) the space of all functions \(f\) defined on \({\mathbb{R}}^d\) having bounded derivatives \(\partial^k f\) for multi-indices \(k\in{\mathbb{N}}^d\) with \(|k|\leqslant\alpha\) so that \[\begin{align} \Vert f\Vert_{\mathcal{C}^\alpha}&\mathrel{\vcenter{:}}=\|f\|_{\mathcal{C}^{ \lfloor\alpha\rfloor}}+[f]_{\mathcal{C}^\alpha}\mathrel{\vcenter{:}}=\sum_{|k|\leqslant\lfloor\alpha\rfloor}\sup_{x\in{\mathbb{R}}^d}|\partial^kf(x)|+\sum_{k=\lfloor\alpha\rfloor} [\partial^kf]_{\mathcal{C}^{\{\alpha\}}}<\infty, \end{align}\] where \(\{\alpha\}:=\alpha-\lfloor\alpha\rfloor\). Note that the \(\mathcal{C}^\alpha\)-norm always includes the supremum of the function. We also denote the space of bounded measurable functions \(\mathcal{C}^0({\mathbb{R}}^d)\) with the supremum norm. To be noticed that the functions in \(\mathcal{C}^0\) do not need to be continuous.
Let \(n\in{\mathbb{N}}\) and \(\alpha,\beta\in(0,1)\) such that \(\alpha+\beta>1\). Then for \(f\in {\mathcal{C}}^\alpha([0,T],{\mathbb{R}}^n)\), \(g\in {\mathcal{C}}^\beta([0,T],{\mathbb{R}}^n)\), the Young integral \(h_t=\int_{0}^{t}f_tdg_t\) is well-defined as the limit as \(m\rightarrow\infty\) of the Riemann sums \[\begin{align} \sum_{i=0}^{m}f_{t_i^m}(g_{t_{i+1}^m\wedge t}-g_{t_{i}^m\wedge t}) \end{align}\] where \((t_i^m)_{0\leqslant m}\) is any partition sequence of \([0,t]\). And the Young integral satisfies the estimates: there exists the constant \(C\) depends only on \(\alpha,\beta\) so that for all \(0\leqslant s\leqslant t\leqslant T,\) \[\begin{align} |h_t-h_s-f_s(g_t-g_s)|\leqslant C|t-s|^{\alpha+\beta}[f]_{{\mathcal{C}}^\alpha([s,t])}[g]_{{\mathcal{C}}^\beta([s,t])}, \end{align}\] which also yields the following \[\begin{align} \label{Young-est} [h]_{{\mathcal{C}}^\beta([s,t])}\leqslant C\Vert f\Vert_{{\mathcal{C}}^\alpha([s,t])}[g]_{{\mathcal{C}}^\beta([s,t])}. \end{align}\tag{6}\]
In the following we denote the conditional expectation w.r.t. the \(\sigma\)-algebras of the filtration \((\mathcal{F}_t)_{t\geqslant 0}\) as \({\mathbb{E}}^t(\cdot)\mathrel{\vcenter{:}}={\mathbb{E}}(\cdot|\mathcal{F}_t), t\geqslant 0\), \(\|X\|_{L^p_\omega}:=({\mathbb{E}}|X|^p)^\frac{1}{p}\), \(\|X\|_{L^p_\omega|{\mathcal{F}}_s}:=({\mathbb{E}}[|X|^p|{\mathcal{F}}_s])^\frac{1}{p}\).
For \(p \in [1,\infty]\), \(X \in L^p(\Omega,{\mathbb{R}}^d)\) and \(\mathcal{F}_s\)-measurable \({\mathbb{R}}^d\) valued random vector \(Y\), we have the following inequalities \[\label{eq:CJI} \|{\mathbb{E}}^s X\|_{L^p_\omega} \leqslant\| X\|_{L^p_\omega}\tag{7}\] and \[\label{eq:condition} \| X-{\mathbb{E}}^s X \|_{L^p_\omega|\mathcal{F}_s} \leqslant 2\| X-Y \|_{L^p_\omega|\mathcal{F}_s}\quad a.s.\tag{8}\] Let \(f:[0,1]\times\Omega\rightarrow \mathbb{R}^d\) be a measurable function adapted to the filtration \((\mathcal{F}_t)_{t\geqslant 0}\), \(\gamma\in(0,1]\), \(p\geqslant 2\) and \([S,T]\subset[0,1]\). We give the following definitions: \[\begin{align} &\| f \|_{C_p^0[S,T]}\mathrel{\vcenter{:}}=\sup_{r \in [S,T]} \| f(r) \|_{L^p_\omega};\\ &[f]_{C_p^{\gamma}[S,T]} \mathrel{\vcenter{:}}= \sup_{r_1, r_2 \in [S,T],r_1\neq r_2} \frac{\| \partial^{\lfloor \gamma \rfloor} f(r_1)-\partial^{\lfloor \gamma \rfloor}f(r_2) \|_{L^p_\omega}}{|r_1-r_2|^{\{\gamma\}}};\\ &\| f \|_{C_p^{\gamma}[S,T]}\mathrel{\vcenter{:}}=\| f \|_{C_p^0[S,T]}+[f]_{C_p^{\gamma}[S,T]}. \end{align}\] If \(f\) is an adapted process, we choose \(Y\) in 8 as the value at \(t\) of the Taylor expansion of \(f\) at \(s\) up to order \(\lfloor \gamma \rfloor\) and we obtain \[\label{eq:fregularity} \| f_t-{\mathbb{E}}^s f_t \|_{L^p_\omega} \leqslant 2|t-s|^{\gamma}[f]_{{C}^{\gamma}_p[s,t]}.\tag{9}\]
In proofs, the notation \(a\lesssim b\) (respectively \(a\asymp b\)) abbreviates the existence of \(C>0\) such that \(a\leqslant C b\) (respectively \(C^{-1}b\leqslant a\leqslant C b\)), such that moreover \(C\) depends only on the parameters claimed in the corresponding statement.
By scaling we can always take \(t\in[0,1]\) without lost of generality. Our main assumption and results are stated as follows.
Assumption 1. Let \(H\in (1,\infty)\backslash{\mathbb{Z}}\), \(b\in {\mathcal{C}}^\alpha\) with \(\alpha\in(1-\frac{1}{2H},1]\).
Notice that following [1], under the assumption above, there exists a unique strong solution to 3 . Here is our numerical approximation result for it.
Theorem 1. Let \((X_t)_{t\in[0,1]}, (X_t^n)_{t\in[0,1]}\) be the solutions to 3 and 5 accordingly. Suppose 1 holds. Then for every \(p\geqslant 1\) and \(\gamma<1+(\alpha-1)H\), we have \[\begin{align} \label{est:strong-main} \| X-X^n \|_{C_p^{\gamma}[0,1]} \leqslant C |x_0-x_0^n| +C n^{-1}, \end{align}\tag{10}\] where \(C=C(p,d,\alpha,H,\|b\|_{{\mathcal{C}}^\alpha})\).
Remark 2. As an easy application of Kolmogorov continuity criteria we can also conclude from 10 the following: \[\begin{align} \big \|\sup_{t\in[0,1]}|X_t-X_t^n|\big\|_{L_\omega^p}\leqslant C |x_0-x_0^n| +C n^{-1}. \end{align}\]
The following theorem settles the question on optimality.
Theorem 3. Let \((X_t)_{t\in[0,1]}, (X_t^n)_{t\in[0,1]}\) be the solutions to 3 and 5 accordingly and \(x_0=x_0^n\). Suppose 1 holds. Then for every \(f\in {\mathcal{C}}^\alpha({\mathbb{R}}^d)\) there exists a \(({\mathcal{F}}_t)_{t\in[0,1]}\)-adapted process \(({\mathcal{H}}f)^X_{t\in[0,1]}\in C_p^{1+(\alpha-1)H-}[0,1]\) for every \(p\geqslant 2\) such that bounded linearly \[\begin{align} {\mathcal{C}}^{\alpha}\ni f\mapsto ({\mathcal{H}}f)^X\in C_p^{1+(\alpha-1)H-}[0,1] \end{align}\] and if \(g\in{\mathcal{C}}^1({\mathbb{R}}^d)\), with probability one \[\begin{align} ({\mathcal{H}}g)_t^X=\int_0^t\nabla g(X_s)\mathrm{d} s,\quad t\in[0,1]; \end{align}\] moreover, we have for any \(t\in[0,1]\), in probability \[\begin{align} \label{eq:optimal} \lim_{n\rightarrow\infty} n(X_t-X_t^n)=:c(t) \end{align}\tag{11}\] exists and \(c\in {\mathcal{C}}^{1+(\alpha-1)H-}([0,1])\) satisfies \[\label{opt-ODE} c(t)= \int_0^tc(s)\mathrm{d} ({\mathcal{H}}b)^X_s+ \frac{1}{2} (b(X_t)-b(x_0)), \quad t\in[0,1],\tag{12}\] where the integral inside 12 is defined in the sense of Young.
The detailed proofs of 1 and 3 will be given in 4 and 5, respectively. Here we only outline the main ideas.
(Convergence) Observe that for any \(p\geqslant 1\), for \(0\leqslant s \leqslant t\leqslant 1\), \[\begin{align} \big \|(X_t-X_t^n)-(X_s-X_s^n)\big\|_{L^p_\omega}= \big\|(\varphi_t-\varphi_t^n)-(\varphi_s-\varphi_s^n)\big\|_{L^p_\omega}, \end{align}\] where \[\begin{align} &\varphi_t:=X_t-B_t^H=\int_0^tb(X_r)\mathrm{d} r=\int_0^tb(\varphi_r+B_r^H)\mathrm{d} r,\\ &\varphi_t^n:=X_t^n-B_t^H=\int_0^tb(X_{k_n(r)}^n)\mathrm{d} r=\int_0^tb(\varphi_{k_n(r)}^n+B_{k_n(r)}^H)\mathrm{d} r, \end{align}\] meanwhile, we know that our aim is to remove the \({\mathcal{C}}^1\) regularity requirement on \(b\) via proper use of the regularization coming from \(B^H\), namely the Gaussian density \(p_{c(H)t^{2H}}\), which is infinitely smoothing (see 3.2). Also keep in mind that in the current setting the Girsanov Theorem is not available. What we have learned from [1], [15] is that we can achieve this goal via freezing the exponent \(\varphi_r\) inside the integral \(\int_0^tb(\varphi_r+B_r^H)\mathrm{d} r\) (and similarly for \(\varphi^n_{k_n(r)}\) inside the integral \(\int_0^tb(\varphi^n_{k_n(r)}+B_r^H)\mathrm{d} r\)) by taking conditional expectation in the framework of the SSL. That is to say, heuristically, for \(t-s\) small enough, \[\begin{align} \int_s^tb(\varphi_r+B_r^H)\mathrm{d} r &\overset{\|\cdot\|_{L^p_\omega}} {\approx}\int_s^t{\mathbb{E}}^{s-(t-s)}b({\mathbb{E}}^{s-(t-s)}\varphi_r+B_r^H)\mathrm{d} r,\\ \int_s^tb(\varphi^n_{k_n(r)}+B_r^H)\mathrm{d} r&\overset{\|\cdot\|_{L^p_\omega}}{\approx} \int_s^t{\mathbb{E}}^{s-(t-s)}b({\mathbb{E}}^{s-(t-s)}\varphi^n_{k_n(r)}+B_r^H)\mathrm{d} r, \end{align}\] and this “\(\overset{\|\cdot\|_{L^p_\omega}} {\approx}\)” is justified by the SSL (see 1 below) by taking \[\begin{align} A_{s,t}:=\int_s^t{\mathbb{E}}^{s-(t-s)}b({\mathbb{E}}^{s-(t-s)}\varphi_r+B_r^H)\mathrm{d} r, \quad {\mathcal{A}}_{s,t}:=\int_s^tb(\varphi_r+B_r^H)\mathrm{d} r, \end{align}\] and similarly for \(\int_s^tb(\varphi^n_{k_n(r)}+B_r^H)\mathrm{d} r\). Then, together with the property of the Gaussian density \(p_{c(H)t^{2H}}\) of fBM ( \(\mathcal{P}_t^Hf:=p_{c(H)t^{2H}}\ast f\)), we can further write \[\begin{align} A_{s,t}= \int_s^t \mathcal{P}^H_{r-[s-(t-s)]} b ({\mathbb{E}}^{s-(t-s)} B^H_r + {\mathbb{E}}^{s-(t-s)} \varphi_r)\mathrm{d} r. \end{align}\] Now we can see that instead of dealing with \(b\) directly, we gain additional regularity in \(\mathcal{P}^H_{t} b\) due to the smoothing effect of the convolution with \(p_{c(H)t^{2H}}\).
Although the full analysis later also contains many technical details, the core of the argument is clear. In the end, we are able to turn the idea above into a proof of the convergence rate in the following form: \[\begin{align} \big\|(X_t-X_t^n)-(X_s-X_s^n)\big\|_{L^p_\omega}=\big\|\varphi_t-\varphi_t^n\big\|_{L^p_\omega} &\leqslant C ( \| \varphi-\varphi^n \|_{C^{\gamma}_p[s,t]} + n^{-1}) |t-s|^{\gamma+\varepsilon}\\ &=C ( \| X-X^n \|_{C^{\gamma}_p[s,t]} + n^{-1}) |t-s|^{\gamma+\varepsilon} \end{align}\] for sufficiently small \(\varepsilon\). Therefore, we obtain 10 after finely dividing the interval \([0,1]\) and applying the above estimate on each sub-interval.
(Optimality) Our idea for verifying optimality is to establish a limit theorem for the asymptotic error distribution of \(X^n\) and \(X\), that is, to show that \(n(X^n-X)\) converges to a nontrivial limit, indicating that order \(n\) is the optimal convergence rate of \(X^n\) from 5 to \(X\).
For this, morally, if the coefficient \(b\) is smooth, we can show that there exists a nonzero process \(c(t)\) such that \(c(t)=\lim_{n\rightarrow\infty} n(X_t-X_t^n)\) in probability and it satisfies a random ODE \[\begin{align} \label{eq:ODE1}c(t)= \int_0^t\nabla b (X_{s})c(s)ds+ \frac{1}{2} (b (X_{t})-b(x_0)). \end{align}\tag{13}\] At this point, we already see the difficulty in our setting: \(b\in {\mathcal{C}}^\alpha\) with \(\alpha\in(1-\frac{1}{2H},1)\) implies that \(\nabla b (X)\) is ill-defined in the above equation.
However, if we reformulate 13 equivalently as follows: denote \(H_t:=\int_0^t \nabla b(X_s)\mathrm{d} s\), \[\begin{align} \label{eq:ODE2} c(t)= \int_0^tc(s)\mathrm{d} H_s+ \frac{1}{2} (b (X_{t})-b(x_0)), \end{align}\tag{14}\] then, due to the regularization effect, we can show that a.s. \(H\) is \({\mathcal{C}}^{1+(\alpha-1)H-}\)-Hölder continuous (see 8), which implies that the term \(\int_0^tc(s)\mathrm{d} H_s\) can be defined in the sense of the Young integral. Therefore, we can claim that 13 is well defined, given that 14 is a linear equation. In this way, we also show that \(n(X^n-X)\) indeed converges to a nontrivial limit and thus that order \(n\) is optimal.
Let us also point out that the general idea of this part is close to that in [18]. However, here we are dealing with additive fBM noise, whereas [18] considers central limit theorem type results that heavily rely on the multiplicative structure of the noise; therefore, the detailed and essential analysis deviates. In the end, unlike the weakly convergence result from [18], we are able to show that \(n(X^n_t-X_t)\) converges to \(c(t)\) in probability.
In this section, we primarily introduce our main tool which is the stochastic sewing lemma and present some properties of the fractional Brownian motions.
Given \(M \geqslant 0\) we define \([S,T]_M^2=\{ (s,t)|S \leqslant s < t \leqslant T, s-M(t-s)\geqslant S \}\) and \(\overline{[S,T]}_M^3=\{ (s,u,t)|(s,t) \in [S,T]_M^2, (u-s)\wedge(t-u)\geqslant\frac{t-s}{3}\}\).
Lemma 1. [1] Let \(0 \leqslant S<T \leqslant 1, p \in[2, \infty), M \geqslant 0\) and let \(\left(A_{s, t}\right)_{(s, t) \in[S, T]_{M}^{2}}\) be a family of random variables in \(L^{p}\left(\Omega, \mathbb{R}^{d}\right)\) such that \(A_{s, t}\) is \(\mathcal{F}_{t}\)-measurable. Suppose that for some \(\varepsilon_{1}, \varepsilon_{2}>0\) and \(C_{1}, C_{2}\) the bounds \[\label{sew:A} \big\|A_{s, t}\big\|_{L^{p}_{\omega}} \leqslant C_{1}|t-s|^{1 / 2+\varepsilon_{1}}\tag{15}\] and \[\label{sew:deltaA} \big\|\mathbf{E}^{s-M(t-s)} \delta A_{s, u, t}\big\|_{L^{p}_{\omega}} \leqslant C_{2}|t-s|^{1+\varepsilon_{2}}\tag{16}\] hold for all \((s, t) \in[S, T]_{M}^{2}\) and \((s, u, t) \in \overline{[S, T]}_{M}^{3}\), where \(\delta A_{s,u,t}:=A_{s,t}-A_{s,u}-A_{u,t}\). Then there exists a unique (up to modification) adapted process \(\mathcal{A}:[S, T] \rightarrow L^{p}\left(\Omega, \mathbb{R}^{d}\right)\) such that \(\mathcal{A}_{S}=0\) and such that for some constants \(K_{1}, K_{2}<\infty\), depending only on \(\varepsilon_{1}, \varepsilon_{2}, p, d\), and \(M\), the bound \[\label{eq:sew:A95bound} \left\|\mathcal{A}_{t}-\mathcal{A}_{s}\right\|_{L^{p}_{\omega}} \leqslant K_{1} C_{1}|t-s|^{1 / 2+\varepsilon_{1}}+K_{2} C_{2}|t-s|^{1+\varepsilon_{2}}\tag{17}\] holds for all \((s, t) \in[S, T]_{0}^{2}\). Moreover, if there exists any continuous process \(\widetilde{\mathcal{A}}\) : \([S, T] \rightarrow L^{p}\left(\Omega, \mathbb{R}^{d}\right), \varepsilon_{3}>0\), and \(K_{3}<\infty\), such that \(\widetilde{\mathcal{A}}_{S}=0\) and \[\label{sew:A-A} \big\|\widetilde{\mathcal{A}}_{t}-\widetilde{\mathcal{A}}_{s}-A_{s, t}\big\|_{L^{p}_{\omega}} \leqslant K_{3}|t-s|^{1+\varepsilon_{3}}\tag{18}\] holds for all \((s, t) \in[S, T]_{M}^{2}\), then \(\widetilde{\mathcal{A}}_{t}=\mathcal{A}_{t}\) for all \(S \leqslant t \leqslant T\).
We recall the following properties concerning \(B^H\) that have been used heavily in later analysis.
Lemma 2. [1] For any \(H \in (0,\infty) \textbackslash \mathbb{Z}\) there exists a constant \(c(H)\) such that for all \(0 \leqslant s \leqslant t \leqslant 1\) one has \[\begin{align} \label{eq:rep1} {\mathbb{E}}|B_t^H-{\mathbb{E}}^s B_t^H |^2=dc(H) |t-s|^{2H} \text{ and } B_t^H-{\mathbb{E}}^s B_t^H \text{ is independent of } \mathcal{F}_s. \end{align}\tag{19}\]
For any \(H \in (0,\infty) \textbackslash \mathbb{Z}\) there exists a constant \(C=C(d,H)\) such that for all \(0 \leqslant s \leqslant t \leqslant 1\) one has \[\begin{align} \label{eq:rep} {\mathbb{E}}|B_t^H- B_s^H | \leqslant C |t-s|^{H \wedge 1}. \end{align}\tag{20}\]
We let \(p_t(x)\) denote the known heat density \(\frac{1}{(2\pi t)^{d/2}}e^{-\frac{|x|^2}{2t}}\) on \({\mathbb{R}}^d\) and we define \({\mathcal{P}}_t^Hf(x):=(p_{c(H)t^{2H}}\ast f)(x)\), \(x\in{\mathbb{R}}^d\). Then for any \(\mathcal{F}_s\)-measurable \({\mathbb{R}}^d\) valued random vector \(\xi\), we have \[\label{eq:EBt} {\mathbb{E}}^sf(B_t^H+\xi)={\mathcal{P}}_{t-s}^H f({\mathbb{E}}^sB_t^H+\xi).\tag{21}\]
Lemma 3. For \(\alpha,\beta \in [0,1],f\in {\mathcal{C}}^\alpha, t \in (0,1],\) one has the bounds, with some constant \(C\) depending only on \(H, \alpha, \beta, d\), accordingly \[\begin{align} \tag{22} & |\mathcal{P}_t^H f(x_1)-\mathcal{P}_t^H f(x_2)-\mathcal{P}_t^H f(x_3)+ \mathcal{P}_t^H f(x_4)| \notag\\ & \leqslant C \|f\|_{{\mathcal{C}}^\alpha}\big( t^{H(\alpha-2)} |x_1-x_2||x_1-x_3| + t^{H(\alpha-1)} |x_1-x_2-x_3+x_4| \big),\forall x_i\in{\mathbb{R}}^d, i=1,\ldots,4; \\ \tag{23} &\|\mathcal{P}_t^H f \|_{{\mathcal{C}}^\beta} \leqslant C t^{H(\alpha-\beta)\wedge0}\|f\|_{{\mathcal{C}}^\alpha};\\ \tag{24} & \|(\mathcal{P}_{t}^H-\mathcal{P}_{s}^H )f \|_{{\mathcal{C}}^\beta} \leqslant C s^{H(\alpha-\beta)-2H\delta} |t^{2H}-s^{2H}|^{\delta}\|f\|_{{\mathcal{C}}^\alpha}, \forall 0\leqslant s\leqslant t\leqslant 1, 0<\delta\in \left[\frac{\alpha-\beta}{2},1 \right]. \end{align}\]
Proof. 22 are directly from [1]. By using properties of Gaussian convolutions, heat kernel bounds and a relation of the form \({\mathcal{P}}_t^Hf(x)=(p_{c(H)t^{2H}}\ast f)(x)\), we get 23 from [14]. For 24 , it holds from [14]. ◻
In this part we give the proof for 1. Denote \[\begin{align} \varphi_t\mathrel{\vcenter{:}}=(X-B^H)_t&=x_0+\int_0^t b(\varphi_s+B^H_s) \mathrm{d}s,\\ \varphi^n_t\mathrel{\vcenter{:}}=(X^n-B^H)_t&=x_0^n+\int_0^t b(\varphi_{k_n(s)}^n+B^H_{k_n(s)}) \mathrm{d} s. \end{align}\] Fix \(S \leqslant s < t \leqslant T\) and \([S,T] \subset [0,1]\). We write \[\begin{align} (X&-X^n)_t-(X-X^n)_s\\&=(\varphi-\varphi^n)_t-(\varphi-\varphi^n)_s \\ &=\int_s^t b(B^H_r+\varphi_r)-b(B^H_r+\varphi_r^n) \mathrm{d} r + \int_s^t b(B^H_r+ \varphi_r^n)-b(B^H_r+ \varphi_{k_n(r)}^n ) \mathrm{d}r \\ & \quad + \int_s^t b(B^H_r+ \varphi^n_{k_n(r)})-b(B^H_{k_n(r)}+\varphi_{k_n(r)}^n) \mathrm{d} r\\ &\eqqcolon \mathcal{E}^{b,n,1}_{s,t}+\mathcal{E}^{b,n,2}_{s,t}+\mathcal{E}^{b,n,3}_{s,t}. \end{align}\] It is clear that in order to show 10 , we need to estimate \({\mathcal{E}}^{b,n,1}_{s,t}, {\mathcal{E}}^{b,n,2}_{s,t}, {\mathcal{E}}^{b,n,3}_{s,t}\) individually. We distribute the estimates for each into 5, 6 and 7 correspondingly.
Before that we first present the following auxiliary lemma for the processes \(\varphi\) and \(\varphi^n\) defined above which will be heavily used in the later proofs.
Lemma 4. Assume 1 holds. Then for all \(t>s\) and \(p\geqslant 1\) we have a.s. \[\begin{align} \tag{25} \| \varphi_t-{\mathbb{E}}^s \varphi_t \|_{L^p_{\omega}|\mathcal{F}_s} & \leqslant C \|b\|_{{\mathcal{C}}^\alpha}|t-s|^{1+\alpha H}; \\ \tag{26} \| \varphi^n_t -{\mathbb{E}}^s \varphi_t^n \|_{L^p_\omega| \mathcal{F}_s} & \leqslant C \|b\|_{{\mathcal{C}}^\alpha}|t-s|^{1+\alpha H} \end{align}\] with some constant \(C=C(p,d,\alpha,H)\).
Proof. We only give the proof of 26 .
For fixed \(s,\) define \(s'\) to be the smallest grid point which is bigger or equal to \(s,\) that is, \(s':=\lceil ns \rceil n^{-1}\). It is crucial to note that \(\varphi_{s'}^n\) is \(\mathcal{F}_s\)-measurable. Suppose 26 holds for some \(m \geqslant 0\) in place of \(1+\alpha H.\) This is certainly true for \(m=0\) thanks to the fact that \(b\) is bounded; we proceed now by induction on \(m.\) If \(s \leqslant t <s',\) then \(\varphi_t^n\) is \(\mathcal{F}_s\)-measurable. Hence \(\varphi^n_t={\mathbb{E}}^s \varphi_t^n\) and the left-hand side of 26 is zero. Therefore it remains to consider the case \(t \geqslant s'.\) In this case, using 8 with \(X=\varphi_t^n, Y=\varphi^n_{s'}+\int_{s'}^{t} b({\mathbb{E}}^s B_{k_n(r)}^H+{\mathbb{E}}^s \varphi_{k_n(r)}^n ) \mathrm{d} r,\) we deduce \[\begin{align} \| \varphi_t^n-{\mathbb{E}}^s \varphi_t^n \|_{L^p_\omega|\mathcal{F}_s} & \leqslant 2 \left\| \varphi_t^n - \varphi^n_{s'}-\int_{s'}^{t} b ({\mathbb{E}}^s B^H_{k_n(r)}+ {\mathbb{E}}^s \varphi_{k_n(r)}^n ) \mathrm{d} r \right\|_{L^p_\omega|\mathcal{F}_s} \\ & =2 \left\| \int_{s'}^t ( b(B^H_{k_n(r)}+\varphi_{k_n(r)}^n)-b({\mathbb{E}}^s B_{k_n(r)}^H+{\mathbb{E}}^s \varphi_{k_n(r)}^n ) \mathrm{d} r \right\|_{L^p_\omega|\mathcal{F}_s} \\ & \leqslant C \|b\|_{{\mathcal{C}}^\alpha}\left\| \int_{s'}^t (|B^H_{k_n(r)}-{\mathbb{E}}^s B^H_{k_n(r)}|^\alpha + | \varphi_{k_n(r)}^n-{\mathbb{E}}^s \varphi^n_{k_n(r)}|^\alpha ) \mathrm{d} r \right\|_{L^p_\omega|\mathcal{F}_s}. \end{align}\] Using 21 and the induction hypothesis, we get \[\| \varphi_t - {\mathbb{E}}\varphi_t \|_{L^p_\omega|\mathcal{F}_s} \leqslant C\|b\|_{{\mathcal{C}}^\alpha} |t-s|^{(H\alpha) \wedge (m \alpha)+1}\quad a.s.\] We note that \(m_0=0, m_{i+1}=1+(H \alpha) \wedge (m_{i} \alpha)\) reaches \(1+H \alpha\) in finitely many steps, therefore we get 25 . ◻
Here we introduce some notations commonly used in the proofs of 5, 6 and 7. For \((s,u,t) \in\overline{[0, 1]}_{1}^{3}\), we set \[\label{eq:points} s_1\mathrel{\vcenter{:}}= s-(t-s), s_2\mathrel{\vcenter{:}}= u-(t-u), s_3\mathrel{\vcenter{:}}= s-(u-s), s_4\mathrel{\vcenter{:}}= s, s_5\mathrel{\vcenter{:}}= u, s_6\mathrel{\vcenter{:}}= t.\tag{27}\] Note by the fact \(u \leqslant\frac{2}{3} s+ \frac{1}{3} t\) for \((s,u, t) \in \overline{[0, 1]}_{1}^{3}\), we have \(s_2 \leqslant s_3.\)
Let us start with the estimate for \(\mathcal{E}^{f,n,1}\).
Lemma 5. Suppose 1 holds. Then for any \(p\geqslant 1\) and \(\gamma\in(\frac{1}{2},1+(\alpha-1)H)\) we have \[\begin{align} \label{est:E1} &\left\| \mathcal{E}^{f,n,1}_{s,t} \right\|_{L^p_\omega} \leqslant C\|f\|_{{\mathcal{C}}^\alpha}\| \varphi-\varphi^n \|_{C_p^{\gamma}[s,t]} |t-s|^{\gamma+\varepsilon}, \quad (s,t)\in[0,1]_0^2,f\in {\mathcal{C}}^{\alpha} \end{align}\tag{28}\] with sufficiently small \(\varepsilon>0\) and some constant \(C=C(p,d,\alpha,\gamma,H,\varepsilon,\|b\|_{{\mathcal{C}}^{\alpha}})\).
Proof. The idea is to apply 1. Let \(M=1\), \((s,t)\in[0,1]_1^2\) and \[A_{s,t}:={\mathbb{E}}^{s-(t-s)} \int_s^t f (B^H_r+ {\mathbb{E}}^{s-(t-s)} \varphi_r)-f(B^H_r+{\mathbb{E}}^{s-(t-s)} \varphi_r^n) \mathrm{d}r.\] We are going to verify 15 and 16 . By 21 , we see \[\begin{align} \label{eq:A95st95heat} A_{s,t} & = \int_s^t \mathcal{P}^H_{r-[s-(t-s)]} f ({\mathbb{E}}^{s-(t-s)} B^H_r + {\mathbb{E}}^{s-(t-s)} \varphi_r) \notag \\ & \qquad -\mathcal{P}^H_{r-[s-(t-s)]} f({\mathbb{E}}^{s-(t-s)} B^H_r+{\mathbb{E}}^{s-(t-s)}\varphi^n_r) \mathrm{d} r. \end{align}\tag{29}\] Then by 23 and 7 , we get \[\label{eq:AstLp} \begin{align} \| A_{s,t} \|_{L^p_\omega} &\leqslant C \|f\|_{{\mathcal{C}}^\alpha} \int_s^t (r-[s-(t-s)])^{-(1-\alpha)H} \| {\mathbb{E}}^{s-(t-s)} (\varphi_r-\varphi_r^n) \|_{L^p_\omega} \mathrm{d}r\\ & \leqslant C \|f\|_{{\mathcal{C}}^\alpha} \| \varphi-\varphi^n \|_{C_p^0[s,t]} |t-s|^{1-(1-\alpha)H}. \end{align}\tag{30}\] Then 15 holds with \(C_1=C \|f\|_{{\mathcal{C}}^\alpha[s,t]} \| \varphi-\varphi^n \|_{C_p^0}\) by the fact that \(1-(1-\alpha)H > \frac{1}{2}\).
Next we verify 16 . Let \((s,u,t)\in\overline{[0,1]}_1^3\). Recall the definition of \(s_i,i=1,\dots,6\) in 27 . We first can write \[\begin{align} &{\mathbb{E}}^{s-(t-s)} \delta A_{s,u,t}\\ =&{\mathbb{E}}^{s_1} {\mathbb{E}}^{s_3} \int_{s_4}^{s_5} f(B_r^H+ {\mathbb{E}}^{s_1} \varphi_r)- f( B_r^H + {\mathbb{E}}^{s_1} \varphi_r^n)-f(B^H_r + {\mathbb{E}}^{s_3} \varphi_r) +f( B^H_r + {\mathbb{E}}^{s_3} \varphi_r^n) \mathrm{d} r \\ &+ {\mathbb{E}}^{s_1} {\mathbb{E}}^{s_2} \int_{s_5}^{s_6} f( B_r^H+{\mathbb{E}}^{s_1} \varphi_r) - f( B_r^H + {\mathbb{E}}^{s_1} \varphi_r^n)- f( B_r^H+{\mathbb{E}}^{s_2} \varphi_r)+f( B_r^H+{\mathbb{E}}^{s_2} \varphi_r^n) \mathrm{d} r \\ =&{\mathbb{E}}^{s_1} \int_{s_4}^{s_5} \mathcal{P}^H_{r-s_3} f({\mathbb{E}}^{s_3} B_r^H+ {\mathbb{E}}^{s_1} \varphi_r)-\mathcal{P}^H_{r-s_3} f({\mathbb{E}}^{s_3} B_r^H+{\mathbb{E}}^{s_1} \varphi_r^n) \\ &\qquad\qquad- \mathcal{P}^H_{r-s_3} f({\mathbb{E}}^{s_3}B^H_r + {\mathbb{E}}^{s_3} \varphi_r)+\mathcal{P}^H_{r-s_3} f({\mathbb{E}}^{s_3} B^H_r + {\mathbb{E}}^{s_3} \varphi_r^n)\mathrm{d} r \\ & + {\mathbb{E}}^{s_1} \int_{s_5}^{s_6} \mathcal{P}_{r-s_2}^H f({\mathbb{E}}^{s_2} B_r^H+{\mathbb{E}}^{s_1} \varphi_r) - \mathcal{P}^H_{r-s_2} f({\mathbb{E}}^{s_2} B_r^H + {\mathbb{E}}^{s_1} \varphi_r^n)\\ &\qquad\qquad- \mathcal{P}^H_{r-s_2} f({\mathbb{E}}^{s_2} B_r^H+{\mathbb{E}}^{s_2} \varphi_r)+ \mathcal{P}^H_{r-s_3}f({\mathbb{E}}^{s_2} B_r^H+{\mathbb{E}}^{s_2} \varphi_r^n)\mathrm{d} r \\ =: & I_1+I_2.\label{def:deltaA} \end{align}\tag{31}\] The two terms are treated in the exactly same way, so we only detail \(I_1\). By 7 and applying 22 with \[\begin{align} x_1={\mathbb{E}}^{s_3} B_r^H+{\mathbb{E}}^{s_1} \varphi_r^n, x_2={\mathbb{E}}^{s_3} B_r^H+ {\mathbb{E}}^{s_1} \varphi_r, x_3={\mathbb{E}}^{s_3} B^H_r + {\mathbb{E}}^{s_3} \varphi_r^n, x_4={\mathbb{E}}^{s_3}B^H_r + {\mathbb{E}}^{s_3} \varphi_r, \end{align}\] we obtain \[\label{eq:I95195bound} \begin{align} \| I_1 \|_{L^p_\omega} & \leqslant C \|f\|_{{\mathcal{C}}^\alpha} \int_{s_4}^{s_5} |r-s_3|^{-(1-\alpha)H} \big\| {\mathbb{E}}^{s_1} | {\mathbb{E}}^{s_3} \varphi_r -{\mathbb{E}}^{s_3} \varphi_r^n-{\mathbb{E}}^{s_1} \varphi_r+{\mathbb{E}}^{s_1} \varphi_r^n | \big\|_{L^p_\omega} \\ & \quad \qquad\qquad\quad+ |r-s_3|^{-(2-\alpha)H} \big\| |{\mathbb{E}}^{s_1} \varphi_r-{\mathbb{E}}^{s_1} \varphi_r^n | \cdot {\mathbb{E}}^{s_1} | {\mathbb{E}}^{s_3} \varphi_r^n-{\mathbb{E}}^{s_1} \varphi_r^n | \big\|_{L^p_\omega} \mathrm{d}r. \end{align}\tag{32}\] By 26 , \[\label{eq:phi95n95reg} {\mathbb{E}}^{s_1} | {\mathbb{E}}^{s_3} \varphi_r^n -{\mathbb{E}}^{s_1} \varphi_r^n | = {\mathbb{E}}^{s_1} | {\mathbb{E}}^{s_3} ( \varphi^n_r-{\mathbb{E}}^{s_1} \varphi_r^n)| \leqslant{\mathbb{E}}^{s_1} | \varphi^n_r-{\mathbb{E}}^{s_1} \varphi_r^n| \leqslant C |r-s_1|^{1+\alpha H}.\tag{33}\] Besides from 7 , we get \[\label{eq:E1E361E1} \begin{align} \big\|{\mathbb{E}}^{s_1} |{\mathbb{E}}^{s_3} \varphi_r-{\mathbb{E}}^{s_3} \varphi_r^n-{\mathbb{E}}^{s_1} \varphi_r+{\mathbb{E}}^{s_1} \varphi_r^n | \big\|_{L^p_\omega} & =\big\|{\mathbb{E}}^{s_1} | {\mathbb{E}}^{s_3} ((\varphi_r-\varphi_r^n)-{\mathbb{E}}^{s_1}(\varphi_r-\varphi_r^n))|\big\|_{L^p_\omega} \\ & \leqslant\big\|{\mathbb{E}}^{s_1} |(\varphi_r-\varphi_r^n)-{\mathbb{E}}^{s_1}(\varphi_r-\varphi_r^n)|\big\|_{L^p_\omega}\\ & \leqslant\| (\varphi_r-\varphi_r^n)-{\mathbb{E}}^{s_1} (\varphi_r-\varphi_r^n) \|_{L^p_\omega}, \end{align}\tag{34}\] meanwhile 9 implies \[\label{eq:phi-phi95n95reg} \big\|{\mathbb{E}}^{s_1} |{\mathbb{E}}^{s_3} \varphi_r-{\mathbb{E}}^{s_3} \varphi_r^n-{\mathbb{E}}^{s_1} \varphi_r+{\mathbb{E}}^{s_1} \varphi_r^n | \big\|_{L^p_\omega} \leqslant C |r-s_1|^{\gamma} [\varphi-\varphi^n]_{C^{\gamma}_p[s,t]},\tag{35}\] clearly from 7 , \[\label{eq:phi-phi95n95bound} \| {\mathbb{E}}^{s_1} (\varphi_r-\varphi_r^n ) \|_{L^p_\omega} \leqslant\| \varphi-\varphi^n \|_{C_p^0[s,t]},\tag{36}\] now plugging 33 , 35 and 36 into 32 , we have \[\label{est:I1} \begin{align} \| I_1 \|_{L^p_\omega} & \leqslant C \|f \|_{{\mathcal{C}}^\alpha} [\varphi-\varphi^n]_{C_p^{\gamma}[s,t]} \int_{s_4}^{s_5} (r-s_1)^{\gamma} (r-s_3)^{-(1-\alpha)H} \mathrm{d} r \\ & \quad + \|f\|_{{\mathcal{C}}^\alpha} \| \varphi-\varphi^n \|_{C_p^0[s,t]} \int_{s_4}^{s_5} (r-s_1)^{1+\alpha H} (r-s_3)^{-(2-\alpha)H} \mathrm{d} r \\ & \leqslant C \|f\|_{{\mathcal{C}}^\alpha} (t-s)^{1+\gamma-(1-\alpha)H} [\varphi-\varphi^n]_{C_p^{\gamma}[s,t]} \\ & \quad + \|b \|_{{\mathcal{C}}^\alpha} (t-s)^{2+\alpha H-(2-\alpha)H} \| \varphi-\varphi^n \|_{C_p^0[s,t]}\\ & \leqslant C \|f\|_{{\mathcal{C}}^\alpha} \| \varphi-\varphi^n \|_{C_p^{\gamma}[s,t]} (t-s)^{(1+\gamma-(1-\alpha)H)\wedge(2+\alpha H-(2-\alpha)H)}. \end{align}\tag{37}\]
The above analysis also implies the same bound on \(I_2\) observing \(I_1\) and \(I_2\) share the same structure.
Noticing 1 implies \((1+\gamma-(1-\alpha)H)\wedge(2+\alpha H-(2-\alpha)H)>1\), we conclude 16 holds with \(C_2= C \|f\|_{{\mathcal{C}}^\alpha} \| \varphi-\varphi^n \|_{C_p^{\gamma}[s,t]}\).
Now we claim that the process \(\mathcal{A}\) in 17 actually is given by \[{\mathcal{A}}_t= \int_0^t f(B^H_r+\varphi_r)-f(B^H_r+\varphi_r^n) \mathrm{d} r.\] To prove this, it suffices to show 18 . By 29 , we write \[\label{con:uniq} \begin{align} {\mathcal{A}}_t-{\mathcal{A}}_s-A_{s,t} &=\int_s^t f(B^H_r+\varphi_r)-\mathcal{P}^H_{r-s_1} f({\mathbb{E}}^{s_1} B_r^H + {\mathbb{E}}^{s_1} \varphi_r) \mathrm{d}r \\ &\quad -\int_s^t f(B_r^H+ \varphi_r^n)-\mathcal{P}^H_{r-s_1} f({\mathbb{E}}^{s_1} B_r^H + {\mathbb{E}}^{s_1} \varphi_r^n) \mathrm{d}r \\ & =:II_1+II_2. \end{align}\tag{38}\] Again we can see that \(II_2\) can be treated similarly to \(II_1\), so we only detail \(II_1\). We can see \[\begin{align} \label{est:pt-12} II_1&=\int_s^t (f-\mathcal{P}^H_{r-s_1} f)(B_r^H+\varphi_r) \mathrm{d}r + \int_s^t \mathcal{P}^H_{r-s_1} f(B_r^H+\varphi_r)-\mathcal{P}^H_{r-s_1} f({\mathbb{E}}^{s_1} B_r^H+{\mathbb{E}}^{s_1} \varphi_r) \mathrm{d}r. \end{align}\tag{39}\]
Using 24 with \(\delta=\frac{\alpha}{2}\), \(\beta=0\) and 23 with \(\beta=\alpha\), we get \[\begin{align} \| II_1 \|_{L^p_\omega} &\leqslant C \int_s^t \| \mathcal{P}^H_{r-s_1}f-f \|_{{\mathcal{C}}^0} + \| \mathcal{P}^H_{r-s_1} f \|_{{\mathcal{C}}^\alpha} ( \|B_r^H-{\mathbb{E}}^{s_1} B^H_r \|^{\alpha}_{L^{\alpha p}_{\omega}}+ \| \varphi_r-{\mathbb{E}}^{s_1} \varphi_r\|^{\alpha}_{L_{\omega}^{\alpha p}}) \mathrm{d} r \nonumber\\ & \leqslant C \|f\|_{{\mathcal{C}}^\alpha} \int_s^t (r-s_1)^{\alpha H} + (|r-s_1|^{\alpha H}+|r-s_1|^{\alpha(1+\alpha H)}) \mathrm{d} r \nonumber\\ & \leqslant C \|f \|_{{\mathcal{C}}^\alpha} (t-s)^{(1+\alpha H) \wedge (1+\alpha (1+\alpha H))},\label{con:uni-1} \end{align}\tag{40}\] where in the second inequality we used 20 and 25 . The same bound on \(II_2\).
Therefore, 18 holds since \((1+\alpha H) \wedge (1+\alpha (1+\alpha H))>1\). Then the uniqueness from 1 verifies the claim.
Finally, by 1, the proof completes. ◻
Let us move to estimate \(\mathcal{E}^{f,n,2}\) term.
Lemma 6. Suppose 1 holds. Then we have for any \(p\geqslant 1\), \(\gamma\in(\frac{1}{2},1+(\alpha-1)H)\) and \((s,t)\in[0,1]_0^2\) \[\label{est:E2} \left\| \mathcal{E}^{f,n,2}_{s,t} \right\|_{L^p_\omega} \leqslant\frac{C}{n} \|f\|_{{\mathcal{C}}^\alpha}|t-s|^{\gamma+ \varepsilon},\quad (s,t)\in[0,1]_0^2,f\in {\mathcal{C}}^{\alpha}\tag{41}\] where \(\epsilon>0\) is sufficiently small and \(C=C(p,d,\alpha,\gamma,H,\varepsilon,\|b\|_{{\mathcal{C}}^{\alpha}})\).
Proof. Again the idea is to apply 1. Let \(M=1\), \((s,t)\in[0,1]_1^2\) and \[\label{eq:A95st2} A_{s,t}:={\mathbb{E}}^{s-(t-s)} \int_s^t f(B_r^H+ {\mathbb{E}}^{s-(t-s)} \varphi_r^n)-f(B_r^H+{\mathbb{E}}^{s-(t-s)} \varphi_{k_n(r)}^n) \mathrm{d}r.\tag{42}\] We may notice that the analysis of showing 30 and estimating 31 also work for 42 . Therefore we omit the detailed proof here for showing the following estimates and present in [app:E2]; in the end we get \[\begin{align} \tag{43} \| A_{s,t} \|_{L^p_\omega}&\leqslant C \|f\|_{{\mathcal{C}}^\alpha} \sup_{r \in [s,t]} \| \varphi_r^n-\varphi_{k_n(r)}^n \|_{L_\omega^p} |t-s|^{1-(1-\alpha)H} \nonumber\\& \leqslant C n^{-1} \|f\|_{{\mathcal{C}}^\alpha} |t-s|^{1-(1-\alpha)H},\\ \| {\mathbb{E}}^{s_1} \delta A_{s,u,t}\|_{L^p_\omega}&\leqslant C \frac{\|f\|_{{\mathcal{C}}^\alpha}}{n} (t-s)^{(1+\alpha-(1-\alpha)H) \wedge (2+(2 \alpha -2)H)}, \tag{44} \end{align}\] recall the definition of \(s_i,i=1,\dots,6\) in 27 .
Then, with taking \(C_1=C\|f\|_{{\mathcal{C}}^\alpha}n^{-1}\), 15 holds by the fact that \(1-(1-\alpha)H > \frac{1}{2}.\) 1 implying \((1+\alpha-(1-\alpha)H) \wedge (2+(2 \alpha -2)H)>1\), 16 holds with \(C_2= \frac{C}{n} \|f\|_{{\mathcal{C}}^\alpha}\)
Similarly to 5, we could verify that the process \(\mathcal{A}\) in 17 is indeed given by \[\mathcal{A}_t= \int_0^t f(B^H_r+\varphi_r^n)-f(B^H_r+\varphi_{k_n(r)}^n) \mathrm{d} r.\]
Now it is the analysis for the last term–\(\mathcal{E}^{f,n,3}\).
Lemma 7. Suppose 1 holds. Then for any \(p\geqslant 1\), \(\gamma\in(\frac{1}{2},1+(\alpha-1)H)\) and \((s,t)\in[0,1]_0^2\), we have \[\label{est:E3} \left\| \mathcal{E}^{f,n,3}_{s,t}\right\|_{L^p_\omega} \leqslant\frac{C}{n} \|f\|_{{\mathcal{C}}^\alpha}|t-s|^{\gamma+ \varepsilon},\quad (s,t)\in[0,1]_0^2,f\in {\mathcal{C}}^{\alpha}\tag{45}\] with sufficiently small \(\varepsilon>0\) and some constant \(C=C(p,d,\alpha,\gamma,H,\varepsilon,\|b\|_{{\mathcal{C}}^\alpha})\).
Proof. In order to apply 1, this time we set \[A_{s,t}={\mathbb{E}}^{s-(t-s)} \int_s^t f (B_r^H+ {\mathbb{E}}^{s-(t-s)} \varphi_{k_n(r)}^n)-f(B_{k_n(r)}^H+{\mathbb{E}}^{s-(t-s)} \varphi_{k_n(r)}^n) \mathrm{d}r.\] When \(|t-s| \leqslant\frac{10}{n}\), by 20 , we have for any \(\epsilon>0\) \[\begin{align} \|A_{s,t} \|_{L^p_\omega} & \leqslant C \|f\|_{{\mathcal{C}}^\alpha} \int_s^t \big\| |B^H_r-B^H_{k_n(r)} |^\alpha \big\|_{L^p_\omega} \mathrm{d} r \leqslant C\|f\|_{{\mathcal{C}}^\alpha} |t-s| \cdot \frac{1}{n^\alpha}\\& \leqslant C \|f\|_{{\mathcal{C}}^\alpha} |t-s|^{\gamma+\varepsilon} \frac{1}{n^{\alpha + 1-\gamma-\varepsilon}}. \end{align}\] Since \(\alpha>\gamma\) we can take such sufficiently small \(\varepsilon>0\) so that \(\epsilon\in(0,\alpha-\gamma)\), which implies \[\label{eq:A95st:t-ssmall} \|A_{s,t} \|_{L^p_\omega} \leqslant\frac{C}{n} \|f\|_{{\mathcal{C}}^\alpha} |t-s|^{\frac{1}{2}+\varepsilon}.\tag{46}\] When \(|t-s| > \frac{10}{n}\), we have \[\begin{align} A_{s,t} & =\int_{s_4}^{s_6} \mathcal{P}^H_{r-s_1} f({\mathbb{E}}^{s_1} B_r^H+ {\mathbb{E}}^{s_1} \varphi_{k_n(r)}^n) -\mathcal{P}^H_{k_n(r)-s_1} f({\mathbb{E}}^{s_1} B_{k_n(r)}^H+ {\mathbb{E}}^{s_1} \varphi_{k_n(r)}^n) \mathrm{d}r \\ &=\int_{s_4}^{s_6} \mathcal{P}^H_{r-s_1} f({\mathbb{E}}^{s_1} B_r^H+{\mathbb{E}}^{s_1} \varphi_{k_n(r)}^n)-\mathcal{P}^H_{r-s_1} f({\mathbb{E}}^{s_1} B_{k_n(r)}^H + {\mathbb{E}}^{s_1} \varphi_{k_n(r)}^n) \\ & \qquad +\mathcal{P}^H_{r-s_1} f({\mathbb{E}}^{s_1} B_{k_n(r)}^H+ {\mathbb{E}}^{s_1} \varphi_{k_n(r)}^n)-\mathcal{P}^H_{k_n(r)-s_1} f({\mathbb{E}}^{s_1} B^H_{k_n(r)}+{\mathbb{E}}^{s_1} \varphi_{k_n(r)}^n) \mathrm{d}r \\ &=:IV_1+IV_2. \end{align}\] For \(IV_1\), by 23 , 7 and 20 , we have \[\begin{align} \| IV_1 \|_{L^p_\omega} &\leqslant\int_{s_4}^{s_6} \| \mathcal{P}^H_{r-s_1} f \|_{{\mathcal{C}}^1} \| {\mathbb{E}}^{s_1} (B_r^H-B_{k_n(r)}^H)\|_{L^p_\omega} \mathrm{d}r \leqslant\frac{C\|f\|_{{\mathcal{C}}^\alpha}}{n} \int_{s_4}^{s_6} (r-s_1)^{-(1-\alpha)H} \mathrm{d}r \nonumber\\ & \leqslant\frac{C}{n} \|f\|_{{\mathcal{C}}^\alpha} |t-s|^{1-(1-\alpha)H}.\label{est:I1-3} \end{align}\tag{47}\] For \(IV_2\), 24 with \(\delta=1\) gives us \[\| \mathcal{P}^H_t f-\mathcal{P}_s^H f\|_{{\mathcal{C}}^0} \leqslant C s^{-(2-\alpha)H} |t^{2H}-s^{2H}| \|f\|_{{\mathcal{C}}^\alpha},\] it implies \[\begin{align} |IV_2| & \leqslant C \|f \|_{{\mathcal{C}}^\alpha} \int_{s_4}^{s_6} (k_n(r)-s_1)^{-(2-\alpha)H} \left( (r-s_1)^{2H} -(k_n(r)-s_1)^{2H} \right) \mathrm{d} r . \end{align}\] Moreover, by \(k_n(r)-s_1 \asymp t-s,\) and \[\big| |r-s_1|^{2H}-|k_n(r)-s_1|^{2H}\big| \leqslant C |r-k_n(r)| |r-s_1|^{2H-1} \leqslant\frac{C}{n} |t-s|^{2H-1},\] we have \[|IV_2| \leqslant\frac{C}{n} \|f\|_{{\mathcal{C}}^\alpha} \int_{s_4}^{s_6} (t-s)^{2H-1-(2-\alpha)H} \mathrm{d} r =\frac{C}{n} \|f\|_{{\mathcal{C}}^\alpha} (t-s)^{\alpha H}.\label{est:I2-3}\tag{48}\] Then 46 togehter with 47 and 48 verifies 15 of 1 with taking \(C_1=\frac{C}{n} \|f\|_{{\mathcal{C}}^\alpha}\), since \(((1-(1-\alpha)H)\wedge(\alpha H\big))>\gamma>\frac{1}{2}\).
Next we verify 16 . Let \((s,u,t)\in\overline{[0,1]}_1^3\). Recall the definition of \(s_i,i=1,\dots,6\) in 27 . Similarly to 31 , we have \[\begin{align} {\mathbb{E}}^{s_1} \delta A_{s,u,t} &={\mathbb{E}}^{s_1} \int_{s_4}^{s_5} \mathcal{P}^H_{r-s_3} f({\mathbb{E}}^{s_3} B_r^H + {\mathbb{E}}^{s_1} \varphi_{k_n(r)}^n)-\mathcal{P}^H_{r-s_3} f({\mathbb{E}}^{s_3} B_{k_n(r)}^H + {\mathbb{E}}^{s_1} \varphi^n_{k_n(r)}) \\ & \qquad -\mathcal{P}_{r-s_3}^H f({\mathbb{E}}^{s_3} B^H_r+ {\mathbb{E}}^{s_3} \varphi_{k_n(r)}^n) + \mathcal{P}^H_{r-s_3} f ({\mathbb{E}}^{s_3} B^H_{k_n(r)} + {\mathbb{E}}^{s_3} \varphi^n_{k_n(r)}) \mathrm{d}r \\ &\quad +{\mathbb{E}}^{s_1} \int_{s_5}^{s_6} \mathcal{P}^H_{r-s_2} f({\mathbb{E}}^{s_2} B^H_r + {\mathbb{E}}^{s_1} \varphi^n_{k_n(r)}) - \mathcal{P}^H_{r-s_2} f({\mathbb{E}}^{s_2} B^H_{k_n(r)} + {\mathbb{E}}^{s_1} \varphi^n_{k_n(r)} )\\ &\qquad - \mathcal{P}^H_{r-s_2} f({\mathbb{E}}^{s_2} B^H_r + {\mathbb{E}}^{s_2} \varphi_{k_n(r)}^n)+\mathcal{P}^H_{r-s_2} f({\mathbb{E}}^{s_2} B^H_{k_n(r)} + {\mathbb{E}}^{s_2} \varphi^n_{k_n(r)} ) \mathrm{d} r \\ & =: V_1+ V_2. \end{align}\] Again we observe that the above two terms can be treated in the exactly same way, so we only detail \(V_1\).
Applying 22 with taking \[\begin{align} &x_1={\mathbb{E}}^{s_3} B_{k_n(r)}^H + {\mathbb{E}}^{s_1} \varphi^n_{k_n(r)}, x_2={\mathbb{E}}^{s_3} B_r^H + {\mathbb{E}}^{s_1} \varphi_{k_n(r)}^n, \\& x_3={\mathbb{E}}^{s_3} B^H_{k_n(r)} + {\mathbb{E}}^{s_3} \varphi^n_{k_n(r)}, x_4= {\mathbb{E}}^{s_3} B^H_r+ {\mathbb{E}}^{s_3} \varphi_{k_n(r)}^n, \end{align}\] we get \[\begin{align} \label{Cor4464-V1} & \| V_1 \|_{L^p_\omega} \nonumber \\&\leqslant C \| f \|_{{\mathcal{C}}^\alpha}\int_{s_4}^{s_5} (r-s_3)^{-H(2-\alpha)} \| {\mathbb{E}}^{s_1} [| {\mathbb{E}}^{s_3}(B_r^H-B_{k_n(r)}^H)| \cdot |{\mathbb{E}}^{s_1} \varphi^n_{k_n(r)}-{\mathbb{E}}^{s_3} \varphi^n_{k_n(r)}|] \|_{L^p_\omega}\mathrm{d} r . \end{align}\tag{49}\] By Cauchy-Schwarz inequality, \[\begin{align} &{\mathbb{E}}^{s_1} [| {\mathbb{E}}^{s_3}(B_r^H-B^H_{k_n(r)}) | \cdot |{\mathbb{E}}^{s_1} \varphi^n_{k_n(r)}-{\mathbb{E}}^{s_3} \varphi^n_{k_n(r)}|] \\ & \leqslant({\mathbb{E}}^{s_1} ({\mathbb{E}}^{s_3} (B_r^H-B^H_{k_n(r)}))^2)^{\frac{1}{2}} ({\mathbb{E}}^{s_1} ( {\mathbb{E}}^{s_1} \varphi_{k_n(r)}^n -{\mathbb{E}}^{s_3} \varphi^n_{k_n(r)})^2 )^{\frac{1}{2}}. \end{align}\] By Jensen inequality, we obtain \[\begin{align} ({\mathbb{E}}^{s_3} (B_r^H-B^H_{k_n(r)}))^2 \leqslant&{\mathbb{E}}^{s_3} (B_r^H-B_{k_n(r)}^H)^2,\\ ( {\mathbb{E}}^{s_1} \varphi^n_{k_n(r)} -{\mathbb{E}}^{s_3} \varphi^n_{k_n(r)} )^2 =& ({\mathbb{E}}^{s_3} ( \varphi^n_{k_n(r)} -{\mathbb{E}}^{s_1} \varphi^n_{k_n(r)}))^2 \leqslant{\mathbb{E}}^{s_3} ( \varphi^n_{k_n(r)} - {\mathbb{E}}^{s_1} \varphi^n_{k_n(r)} )^2. \end{align}\] Therefore, by 26 , we have \[\begin{align} &{\mathbb{E}}^{s_1} [| {\mathbb{E}}^{s_3}(B_r^H-B^H_{k_n(r)}) | \cdot |{\mathbb{E}}^{s_1} \varphi^n_{k_n(r)}-{\mathbb{E}}^{s_3} \varphi^n_{k_n(r)}|] \\ & \leqslant( {\mathbb{E}}^{s_1} (B^H_r-B^H_{k_n(r)})^2)^{\frac{1}{2}} \cdot ({\mathbb{E}}^{s_1} (\varphi^n_{k_n(r)} -{\mathbb{E}}^{s_1} \varphi^n_{k_n(r)})^2)^{\frac{1}{2}} \\ & \leqslant C ({\mathbb{E}}^{s_1} (B^H_r-B^H_{k_n(r)})^2)^{\frac{1}{2}} \cdot (k_n(r)-s_1)^{1+\alpha H}. \end{align}\] For \(p \geqslant 2\), 7 and 20 imply \[\label{Cor4464-p622} \begin{align} & \| {\mathbb{E}}^{s_1} [| {\mathbb{E}}^{s_3} (B_r^H-B^H_{k_n(r)})| | {\mathbb{E}}^{s_1} \varphi^n_{k_n(r)}-{\mathbb{E}}^{s_3} \varphi^n_{k_n(r)}|] \|_{L^p_\omega} \\ & \leqslant C (r-s_1)^{1+\alpha H} \left\| \| B_r^H-B^H_{k_n(r)} \|_{L_\omega^2 | \mathcal{F}_{s_1}} \right\|_{L^p_\omega} \\ & = C (r-s_1)^{1+\alpha H} \| {\mathbb{E}}^{s_1} | B_r^H- B_{k_n(r)}^H|^2 \|^{\frac{1}{2}}_{L_\omega^{\frac{p}{2}}}\\ & \leqslant C (r-s_1)^{1+\alpha H} \| B_r^H -B^H_{k_n(r)} \|_{L^p_\omega} \\ & \leqslant\frac{C}{n} (r-s_1)^{1+\alpha H} . \end{align}\tag{50}\] Therefore, plugging 50 into 49 , we have \[\begin{align} \|V_1 \|_{L^p_\omega} & \leqslant\frac{C}{n} \|b\|_{{\mathcal{C}}^\alpha} \int_{s_4}^{s_5} (r-s_3)^{-(2-\alpha)H} (r-s_1)^{1+\alpha H} \mathrm{d} r \leqslant\frac{C}{n} \|f\|_{{\mathcal{C}}^\alpha} |t-s|^{2+(2\alpha-2)H}. \end{align}\] The same bound holds on \(V_2\).
Since 1 implies \(2+(2 \alpha -2)H>1\), we can obtain that 16 holds with \(C_2= \frac{C}{n} \|b\|_{{\mathcal{C}}^\alpha}\).
Similarly to 5, we could verify the process \(\mathcal{A}\) in 17 is given by \[\mathcal{A}_t=\int_0^t f(B^H_r+ \varphi^n_{k_n(r)})-f(B^H_{k_n(r)}+\varphi_{k_n(r)}^n) \mathrm{d} r.\]
In the end all of the conditions from 1 are verified, which proves the desired result. ◻
With 5, 6 and 7 at hand we are ready to give:
Proof of 1. By 28 , 41 and 45 , we see that \[\begin{align} \| ( X-X^n)_t-(X-X^n)_s \|_{L^p_\omega} &\leqslant C ( \| \varphi-\varphi^n \|_{C^{\gamma}_p[s,t]} + n^{-1}) |t-s|^{\gamma+\varepsilon}\\& = C ( \| X-X^n \|_{C^{\gamma}_p[s,t]} + n^{-1}) |t-s|^{\gamma+\varepsilon}, \end{align}\] which implies \[\begin{align} [X-X^n]_{C_p^{\gamma}[S,T]} \leqslant C (\| X-X^n \|_{C_p^{\gamma}[S,T]} + n^{-1} ) (T-S)^\varepsilon. \end{align}\] Therefore, we have \[\begin{align} \| X-X^n \|_{C_p^{\gamma}[S,T]} & \leqslant| (X-X^n)_S| + 2[X-X^n]_{C_p^{\gamma}[S,T]} \\ & \leqslant| (X-X^n)_S| +C (\| X-X^n \|_{C_p^{\gamma}[S,T]} + n^{-1} ) (T-S)^\varepsilon. \end{align}\] Fix \(T-S=\Delta\) small enough and we obtain \[\|X-X^n \|_{C^{\gamma}_p[S,T]} \leqslant C (|(X-X^n)_S| + n^{-1}).\] Dividing \([0,1]\) into \([0, \Delta], [\Delta, 2 \Delta], \ldots\), yields that \[\|X-X^n \|_{C^{\gamma}_p[0,1]} \leqslant C(|x_0-x_0^n|+ n^{-1}).\] ◻
In this section, we discuss the optimality of the convergence rate \(n^{-1}\) obtained in the previous part. By showing that \(n(X-X^n)\) converges to a non-zero limit, we verify that the rate \(n^{-1}\) is optimal for the scheme 5 .
In the following first we present a regularization lemma concerning the solution to 3 .
Lemma 8. Let \((X_t)_{t\in[0,1]}\) be the solution to 3 . Suppose 1 holds. Then for every \(p>1\), for any \(f\in {\mathcal{C}}^1\), we have \[\label{est:reg} \Big\| \int_0^{\cdot} \nabla f(X_t) \mathrm{d} t \Big\|_{C_p^{1+(\alpha-1)H}[0,1]} \leqslant C\|f\|_{{\mathcal{C}}^\alpha},\tag{51}\] where \(C\) depends only on \(d,p,H,\alpha, \|b\|_{{\mathcal{C}}^\alpha}\).
Proof. In order to show 51 , we apply 1. Let \(M=1\), \((s,t)\in[0,1]_1^2\) and \[A_{s,t}:={\mathbb{E}}^{s-(t-s)} \int_s^t \nabla f(B_r^H+ {\mathbb{E}}^{s-(t-s)} \varphi_r) \mathrm{d} r \text{ where } \varphi:=X-B^H.\] We are going to verify 15 and 16 . By 21 , we see \[A_{s,t} = \int_s^t \mathcal{P}^H_{r-[s-(t-s)]} (\nabla f)({\mathbb{E}}^{s-(t-s)} B_r^H + {\mathbb{E}}^{s-(t-s)} \varphi_r) \mathrm{d} r.\] Then by 23 , we have \[\begin{align} |A_{s,t}| \leqslant\int_s^t \| \mathcal{P}^H_{r-[s-(t-s)]} (\nabla f) \|_{C^0} \mathrm{d}r \leqslant C \int_s^t \left[ r-[s-(t-s)] \right]^{H(\alpha-1)} \| f \|_{C^{\alpha}} \mathrm{d} r. \end{align}\] Therefore, we have \(\|A_{s,t}\|_{L^p_\omega} \leqslant C \|f\|_{C^\alpha} (t-s)^{1-(1-\alpha)H }.\) Then 15 holds with \(C_1=C \|f\|_{C^\alpha}\) by the fact that \(1-(1-\alpha)H > \frac{1}{2}\).
Next we verify 16 . Let \((s,u,t)\in\overline{[0,1]}_1^3\). Recall the definition of \(s_i,i=1,\dots,6\) in 27 . We first can write \[\begin{align} {\mathbb{E}}^{s_1} \delta A_{s,u,t} & ={\mathbb{E}}^{s_1} \bigg[ \int_{s_4}^{s_5} {\mathbb{E}}^{s_1} \nabla f(B_r^H+ {\mathbb{E}}^{s_1} \varphi_r)-{\mathbb{E}}^{s_3} \nabla f(B_r^H+ {\mathbb{E}}^{s_3} \varphi_r) \mathrm{d}r \bigg] \\ & \quad + {\mathbb{E}}^{s_1} \bigg[ \int_{s_5}^{s_6} {\mathbb{E}}^{s_1} \nabla f(B_r^H+ {\mathbb{E}}^{s_1} \varphi_r) -{\mathbb{E}}^{s_2} \nabla f(B_r^H+ {\mathbb{E}}^{s_2} \varphi_r ) \mathrm{d} r\bigg] \\ & = \int_{s_4}^{s_5} {\mathbb{E}}^{s_1} \bigg[ \mathcal{P}^H_{r-s_3} \nabla f({\mathbb{E}}^{s_3} B_r^H+ {\mathbb{E}}^{s_1} \varphi_r) - \mathcal{P}^H_{r-s_3} \nabla f ({\mathbb{E}}^{s_3} B_r^H + {\mathbb{E}}^{s_3} \varphi_r)\bigg] \mathrm{d} r \\ & \quad + \int_{s_5}^{s_6} {\mathbb{E}}^{s_1} \bigg[ \mathcal{P}^H_{r-s_2} \nabla f ({\mathbb{E}}^{s_2} B_r^H+ {\mathbb{E}}^{s_1} \varphi_r) - \mathcal{P}^H_{r-s_2} \nabla f ({\mathbb{E}}^{s_2} B_r^H+ {\mathbb{E}}^{s_2} \varphi_r) \bigg] \mathrm{d}r. \end{align}\] By 23 and 25 , we have \[\begin{align} \|{\mathbb{E}}^{s_1} \delta A_{s,u,t}\|_{L^p_\omega} & \leqslant C \int_s^t (r-s_3)^{H(\alpha-2)} \| f \|_{{\mathcal{C}}^{\alpha}}\|{\mathbb{E}}^{s_1} [|{\mathbb{E}}^{s_1} \varphi_r-{\mathbb{E}}^{s_3} \varphi_r|]\|_{L^p_\omega} \mathrm{d}r \\ & \quad + \int_u^t (r-s_2)^{H(\alpha-2)} \| f \|_{{\mathcal{C}}^{\alpha}} \|{\mathbb{E}}^{s_1} [|{\mathbb{E}}^{s_1} \varphi_r-{\mathbb{E}}^{s_2} \varphi_r|]\|_{L^p_\omega} \mathrm{d}r\\ &\leqslant C \|f\|_{{\mathcal{C}}^\alpha} |t-s|^{2+2\alpha H-2H}. \end{align}\] Noticing 1 implies \(2+2\alpha H-2H>1\), we conclude 16 holds with \(C_2= C \|f\|_{{\mathcal{C}}^\alpha}\). Moreover, since \(f\in C^1\), it is obviously to see that the process \(\mathcal{A}\) in 17 actually is given by \[{\mathcal{A}}_t= \int_0^t \nabla f(X_r) \mathrm{d} r.\] Therefore, by 1, we have \[\big\| \int_s^t \nabla f(X_r) \mathrm{d} r \big\|_{L^p} \leqslant C\|f\|_{{\mathcal{C}}^\alpha}|t-s|^{H(\alpha-1)+1},\] which implies that 51 holds. ◻
Due to 51 , we can already state a result which is analogous to [18].
Proposition 4. Let \((X_t)_{t\in[0,1]}\) be the solutions to 3 . Suppose 1 holds. For every \(f\in {\mathcal{C}}^\alpha({\mathbb{R}}^d)\) there exists a \(({\mathcal{F}}_t)_{t\in[0,1]}\)-adapted process \(({\mathcal{H}}f)^X_{t\in[0,1]}\in C_p^{1+(\alpha-1)H-}[0,1]\) for every \(p\geqslant 2\) such that for \(g\in {\mathcal{C}}^1\) with probability one, \[\begin{align} ({\mathcal{H}}g)^X_t=\int_0^t\nabla g(X_s)\mathrm{d} s,\quad t\in[0,1]. \end{align}\]
Proof. In fact, we notice that for any \(f \in {\mathcal{C}}^{\alpha},\) there exists sequence \((f_n)_{n \in \mathbb{N}} \subset {\mathcal{C}}^1\), such that \(\lim_{n \to \infty} f_n=f\) in \({\mathcal{C}}^{\alpha-}.\) Therefore, by 8, we have \(\big( \int_0^{\cdot} \nabla f_n(X_t) \mathrm{d}t \big)_{n \in \mathbb{N}}\) is a Cauchy sequence in \(C_p^{1+(\alpha-1)H-}[0,1].\) We define \({\mathcal{H}}f=\lim_{n\rightarrow\infty}\int_0^{\cdot} \nabla f_n(X_t) \mathrm{d}t\) in \(C_p^{1+(\alpha-1)H-}[0,1].\) ◻
Therefore, for \(b \in {\mathcal{C}}^{\alpha}\), we can define \(\int_0^{\cdot} \nabla b(X_t) \mathrm{d} t:=\lim_{n \to \infty} \int_0^{\cdot} \nabla b_n(X_t) \mathrm{d} t\) in probability. Moreover from Kolmogorov continuity criteria we know that it has a version which has \(\theta\)-Hölder continuous path for \(\theta\in(0, 1+(\alpha-1)H)\subset(0,1)\).
Now we are at the position of showing 3.
Proof of 3. Clearly 4 shows the existence of \(({\mathcal{H}}f)^X\in C_p^{1+(\alpha-1)H-}[0,1]\) which is bounded linear for \(f\in {\mathcal{C}}^{\alpha}\), it also implies that the \(c\) satisfying 12 for given \(X\) is well-defined and \(\|c\|_{{\mathcal{C}}^{1+(\alpha-1)H-}([0,1])}<\infty,{\mathbb{P}}\)-a.s. Indeed from Kolmogorov continuity criteria and the property of Young integral we know that for any possible solution \(c\), for any \(k_1\in(\frac{1}{2},{1+(\alpha-1)H}),k_2\in(\frac{1}{2},(1+(\alpha-1)H)\wedge\alpha)=(\frac{1}{2},{1+(\alpha-1)H})\) so that \(k_1+k_2>1\), for any \(0\leqslant s\leqslant t\leqslant 1\) and \(q>p\), we have \[\begin{align} \label{reg:c} \|c(t)-c(s)\|_{L^p_\omega}\leqslant&\| \int_s^tc(r)\mathrm{d}({\mathcal{H}}b)_r^{X}\|_{L^p_\omega}+\frac{1}{2}\|b(X_t)-b(X_s)\|_{L^p_\omega} \nonumber\\\leqslant C&\big\|\|c\|_{{\mathcal{C}}^{k_1}([0,1])}\|({\mathcal{H}}b)^{ X}\|_{{\mathcal{C}}^{k_2}([0,1])}\big\|_{L^p_\omega}|t-s|^{k_2}+C\|b\|_{{\mathcal{C}}^\alpha({\mathbb{R}}^d)}|t-s|^{\alpha} \nonumber\\\leqslant C&\|c\|_{L_\omega^q({\mathcal{C}}^{k_1}([0,1]))}\|({\mathcal{H}}b)^{ X}\|_{L_\omega^{\frac{qp}{q-p}}({\mathcal{C}}^{k_2}([0,1]))}|t-s|^{k_2}+C\|b\|_{{\mathcal{C}}^\alpha({\mathbb{R}}^d)}|t-s|^{k_2} \nonumber \\\leqslant C&\|b\|_{{\mathcal{C}}^\alpha({\mathbb{R}}^d)}|t-s|^{k_2}. \end{align}\tag{52}\] Then, by Kolmogorov continuity criteria again we get \({\mathbb{P}}\)-a.s. \(\|c\|_{{\mathcal{C}}^{1+(\alpha-1)H-}}<\infty\). Notice that 12 is a linear ODE in the sense of Young integral for given \(X\), standard fixed point arguments gives us the existence and uniqueness of \(c\in {\mathcal{C}}^{1+(\alpha-1)H-}([0,1])\) satisfying 12 .
Therefore we only need to show that 11 holds for such \(c\) and for \(b\in{\mathcal{C}}^\alpha\), \(\alpha<1\).
Denote \(c^n:=n(X-X^n)\). 1 implies that \(\sup_n \|c_n\|_{{\mathcal{C}}_p^{1+(\alpha-1)H-}[0,1]}<\infty,\) that is to say, for \(\gamma<1+(\alpha-1)H\), from Kolmogorov continuity criteria, \(\sup_n \|c_n\|_{L^p(\Omega,{\mathcal{C}}^{\gamma})}<\infty\), hence for \({\mathcal{L}}^n:=(X,X^n,c^n, c, c^n-c, B^H)\), \(\sup_n \|{\mathcal{L}}^n\|_{L^p(\Omega,({\mathcal{C}}^{\gamma})^6)}<\infty.\)
By Sobolve embedding we know that there exists large \(q\) and \(\gamma_1<\gamma_2<\gamma\) so that compactly \[\begin{align} \label{emd:com}{\mathcal{C}}^{\gamma}([0,1],{\mathbb{R}}^d)\subset\subset W_q^{\gamma_2}([0,1],{\mathbb{R}}^d)\subset \subset{\mathcal{C}}^{\gamma_1}([0,1],{\mathbb{R}}^d), \end{align}\tag{53}\] hence the laws of \(({\mathcal{L}}^n)_{n\in{\mathbb{N}}}\) are tight on \(W_q^{\gamma_2}([0,1],{\mathbb{R}}^d)^6\) which is Polish and \[\begin{align} \label{uniform} \sup_n\||{\mathcal{L}}^n|\|_{L^p(\Omega, W_q^{\gamma_2}([0,1],{\mathbb{R}}^d)^6)}<\infty. \end{align}\tag{54}\] Following from Prokorov’s theorem, we have that the laws of a further subsequence \(({\mathcal{L}}^n)_{n\in{\mathbb{N}}_1}, {\mathbb{N}}_1\subset{\mathbb{N}}\), converges weakly on \(W_q^{\gamma_2}([0,1],{\mathbb{R}}^d)^6\). By Skorohod’s representation theorem, there exists a probability space \((\hat{\Omega},\hat{{\mathcal{F}}}, \hat{\mathbb{P}})\) and random variables \(\hat{\mathcal{L}}^n:=(\tilde{X}^n,\hat{X}^{n,n}, \hat{c}^n, \tilde{c}^n,\hat{c}^n-\tilde{c}^n,\hat{B}^{H,n}):\hat{\Omega}\mapsto W_q^{\gamma_2}([0,1],{\mathbb{R}}^d)^6\) and \({\mathcal{L}}:=(\tilde{X},\hat{X}, \hat{c}, \tilde{c}, \hat{c}-\tilde{c}, \hat{B}^H):\hat{\Omega}\mapsto W_q^{\gamma_2}([0,1],{\mathbb{R}}^d)^6\) so that \[\begin{align} \label{eq:samelaw} {\mathcal{L}}^n \text{ on } (\Omega,{\mathcal{F}},{\mathbb{P}})\overset{d}{=}\hat{\mathcal{L}}^n \text{ on }(\hat{\Omega},\hat{{\mathcal{F}}}, \hat{\mathbb{P}}), \end{align}\tag{55}\] and \[\begin{align} \label{con:p-a46s46}\hat{\mathbb{P}}-a.s.,\quad (\hat{\mathcal{L}}^n)_{n\in{\mathbb{N}}_1} \quad \text{converges to } \quad {\mathcal{L}}\quad\text{ in } W_q^{\gamma_2}([0,1],{\mathbb{R}}^d)^6. \end{align}\tag{56}\] Let \((\hat{\mathcal{F}}_t)_{t\geqslant 0}\) be the augmentation of the filtration generated by \((\hat{X}, \hat{X}^n, \hat{B}^{H,n})\). 55 , the continuity of \(b\) and the strong well-posedness of 3 imply \[\begin{align} \tilde{X}_t&=x_0+\int_0^tb(\tilde{X}_s)\mathrm{d}s+\hat{B}_t^H, \quad \tilde{X}_t^k=x_0+\int_0^tb(\tilde{X}_s^k)\mathrm{d}s+\hat{B}_t^{H,k},\\ \hat{X}_t^{k,n}&=x_0+\int_0^tb(\hat{X}_{k_n(s)}^{k,n})\mathrm{d}s+\hat{B}_t^{H,k},\quad \hat{c}^n=n(\tilde{X}^n- \hat{X}^{n,n}), \\ \tilde{c}^n(t)&= \int_0^t\tilde{c}^n(s)\mathrm{d} ({\mathcal{H}}b)^{\tilde{X}_s^n}+ \frac{1}{2} (b(\tilde{X}_t^n)-b(x_0)), \quad t\in[0,1]. \end{align}\] Following from 55 which implies the uniform integrability, 56 and Vitali’s theorem, for all \(p\geqslant 1\) we have \[\begin{align} \label{con:L-Ln} \lim_{n\rightarrow\infty}\big\|\hat{\mathcal{L}}^n-{\mathcal{L}}\big\|_{L^p(\hat{\Omega}, W_q^{\gamma_2}([0,1],{\mathbb{R}}^d)^6)}=0. \end{align}\tag{57}\] At this point we can also claim that \(\tilde{X}=\hat{X}\) \(\hat{\mathbb{P}}\)-a.s. from 1 and the strong well-posedness of 3 .
Notice that for any \(b \in {\mathcal{C}}^{\alpha}\), there exists a sequence \(\{b_n\}_{n \in \mathbb{N}} \subset {\mathcal{C}}^1\) such that \(\lim_{n \to \infty} b_n=b\) in \({\mathcal{C}}^{\alpha-}\), then for \(m\in{\mathbb{N}}\) \[\begin{align} \hat{c}^n(t)=& n(\tilde{X}_t^n-\hat{X}_t^{n,n})= n \left( \int_0^t b( \tilde{X}_s^n) \mathrm{d}s-\int_0^t b( \hat{X}_{k_n(s)}^{n,n}) \mathrm{d}s \right)\nonumber\\=& n \int_0^t (b-b_m)( \tilde{X}_s^n) - (b-b_m)( \hat{X}_{k_n(s)}^{n,n})\mathrm{d}s +n\int_0^tb_m( \tilde{X}_r^n)-b_m( \hat{X}_r^{n,n})\mathrm{d}s\nonumber\\&+n\int_0^tb_m( \hat{X}_r^{n,n})-b_m( \hat{X}_{k_n(s)}^{n,n})\mathrm{d}s=: J_t^{m,n}+K_t^{m,n}+L_t^{m,n}.\label{def:Vn} \end{align}\tag{58}\] Following from 5, 6 and 7 we get \[\begin{align} \label{est:J} \big\|\sup_{t\in[0,1]}|J_t^{m,n}|\big\|_{L_\omega^p}\lesssim\|b-b_m\|_{{\mathcal{C}}^{\alpha-}}; \end{align}\tag{59}\] for \(K^{m,n}\) \[\begin{align} K_t^{m,n}=\int_0^t\int_0^1\nabla b_m(\theta \tilde{X}_r^{n}+(1-\theta) \hat{X}_r^{n,n})\cdot \hat{c}_r^n\mathrm{d}\theta\mathrm{d}r , \end{align}\] since \(\hat{\mathbb{P}}\)-a.s. \(\hat{c}^n\) converges to \(\hat{c}\), \(\hat{X}^{n,n}\) converges to \(\hat{X}\), and \(\tilde{X}^n\) converges to \(\tilde{X}=\hat{X}\), \(\hat{\mathbb{P}}\)-a.s. we have \[\begin{align} \label{est:K} \lim_{n\rightarrow\infty} K_t^{m,n}= \int_0^t\nabla b_m(\hat{X}_r)\cdot \hat{c}_r\mathrm{d}r= \int_0^t \hat{c}_r\mathrm{d}({\mathcal{H}}b_m)_r^{\hat{X}},\quad \hat{\mathbb{P}}-a.s. \end{align}\tag{60}\] Concerning \(L^{m,n}\), following from 5 and the fact a.s. \(B^H\in {\mathcal{C}}^{H-\varepsilon}([0,1])\) for sufficiently small \(\varepsilon>0\) by Kolmogorov continuity criteria, which means there exists a small enough \(\varepsilon'>0\) so that a.s. \(B_H\in{\mathcal{C}}^{1+\varepsilon'}\), we have a.s. (denote \(\{c\}:=c-\lfloor c\rfloor\) for \(c\in{\mathbb{R}}_+\)) \[\begin{align} n(\hat{X}_{r}^n-\hat{X}_{k_ n(r)}^{n,n})&=n(r-k_ n(r))b(\hat{X}_{k_ n(r)}^{n,n})+n(\hat{B}_r^{H,n}-\hat{B}_{k_ n(r)}^{H,n})\\ &=\{nr\}b(\hat{X}_{k_ n(r)}^{n,n})+\{nr\}(\hat{B}_r^{H,n})'+o(n^{-\varepsilon'}); \end{align}\] together with the condition that \(\nabla b_m\) is continuous and bounded, moreover \(\hat{X}_{r}^{n,n}\rightarrow \hat{X}_{r}\) and \(\hat{X}_{k_n(r)}^{n,n}\rightarrow \hat{X}_{r}\) a.s. from 1, dominated convergence theorem shows as \(n\rightarrow\infty\), we have \(\hat{\mathbb{P}}\)-a.s. \[\begin{align} \label{est:L} &\lim_{n\rightarrow \infty}L_t^{m,n} \nonumber\\&= \lim_{n\rightarrow \infty}n\int_0^t\int_0^1\nabla b_m(\theta \hat{X}_r^{n,n}+(1-\theta) \hat{X}_{k_n(r)}^{n,n})\cdot ( \hat{X}_r^{n,n}- \hat{X}_{k_n(r)}^{n,n})\mathrm{d}\theta\mathrm{d}r \nonumber\\&=\lim_{n\rightarrow \infty}\int_0^t\int_0^1\nabla b_m(\theta \hat{X}_r^{n,n}+(1-\theta) \hat{X}_{k_n(r)}^{n,n})\Big(\{nr\}\big(b(\hat{X}_{k_ n(r)}^{n,n})+(B_r^H)'\big)+o(n^{-\varepsilon'})\Big)\mathrm{d}r \nonumber\\&=\frac{1}{2}\int_0^t\nabla b_m(\hat{X}_r)\big(b( \hat{X}_r)+(\hat{B}_r^H)'\big)\mathrm{d}r= \frac{1}{2}\big(b_m(\hat{X}_t)-b_m(x_0)\big), \end{align}\tag{61}\] we have applied the fact that \(\{n\cdot\}\) converges to \(\frac{1}{2}\) weakly in \(L^2([0,t])\) as \(n\rightarrow\infty\) in deriving the penultimate equality and the lase equality holds due to elementary chain rule.
Now we put 59 , 60 and 61 together with taking \(m,n\rightarrow\infty\) for both side of 58 then get \(\hat{\mathbb{P}}\)-a.s. \[\begin{align} \hat{c}(t) =&\int_0^t \hat{c}_r\mathrm{d}({\mathcal{H}}b)_r^{\hat{X}} +\frac{1}{2}(b(\hat{X}_t)-b(x_0)). \end{align}\] That is, \((\hat{X}, \hat{c})\) satisfy 3 and 12 .
Meanwhile following from the fact that \(\hat{\mathbb{P}}\) a.s., \(\tilde{X}^n\) converges to \(\hat{X}\), \(\tilde{c}^n\) converges to \(\tilde{c}\) and the property of Young’s integral 6 we can conclude \(\tilde{c}\) also satisfies 12 for such \(\hat{X}\). It implies that \(\hat{c} =\tilde{c}\) a.s. and \(\hat{\mathbb{P}}\)-a.s., \(\hat{c}^n-\tilde{c}^n\) converges to \(0\); hence we get that for given \(X\), \(c^n\) converges to \(c\) in probability on \((\Omega,{\mathcal{F}},{\mathbb{P}})\). ◻
Proof of 43 and 44 . Similarly to 29 and 30 , we have \[\begin{align} \| A_{s,t} \|_{L^p_\omega}&= \| \int_s^t \mathcal{P}^H_{r-[s-(t-s)]} f ({\mathbb{E}}^{s-(t-s)} B^H_r + {\mathbb{E}}^{s-(t-s)} \varphi_{r}^n) \nonumber\\ & \qquad -\mathcal{P}^H_{r-[s-(t-s)]} f({\mathbb{E}}^{s-(t-s)} B^H_r+{\mathbb{E}}^{s-(t-s)}\varphi^n_{k_n(r)}) \mathrm{d} r\|_{L^p_\omega} \nonumber \\& \leqslant C \|f\|_{{\mathcal{C}}^\alpha} \sup_{r \in [s,t]} \| \varphi_r^n-\varphi_{k_n(r)}^n \|_{L_\omega^p} |t-s|^{1-(1-\alpha)H} \nonumber\\ & \leqslant C n^{-1} \|f\|_{{\mathcal{C}}^\alpha} |t-s|^{1-(1-\alpha)H} ,\label{est:A-2} \end{align}\tag{62}\] where in the second inequality we used \(\| \varphi_r^n-\varphi_{k_n(r)}^n \|_{L_\omega^p}\leqslant\|b\|_{{\mathcal{C}}^0} n^{-1}\). Hence 43 holds.
Next we verify 44 . Let \((s,u,t)\in\overline{[0,1]}_1^3\). Recall the definition of \(s_i,i=1,\dots,6\) in 27 . Similarly to 31 , we can write \[\begin{align} {\mathbb{E}}^{s_1} \delta A_{s,u,t} =&{\mathbb{E}}^{s_1} \int_{s_4}^{s_5} \mathcal{P}_{r-s_3}^H f({\mathbb{E}}^{s_3} B_r^H + {\mathbb{E}}^{s_1} \varphi_r^n)- \mathcal{P}_{r-s_3}^H f({\mathbb{E}}^{s_3} B_r^H + {\mathbb{E}}^{s_1} \varphi_{k_n(r)}^n) \\ & \qquad\qquad- \mathcal{P}^H_{r-s_3} f({\mathbb{E}}^{s_3} B_r^H + {\mathbb{E}}^{s_3} \varphi_r^n)+\mathcal{P}_{r-s_3}^H f({\mathbb{E}}^{s_3} B_r^H+ {\mathbb{E}}^{s_3} \varphi_{k_n(r)}^n)\mathrm{d} r \\ & +{\mathbb{E}}^{s_1} \int_{s_5}^{s_6} \mathcal{P}_{r-s_2}^H f({\mathbb{E}}^{s_2} B_r^H + {\mathbb{E}}^{s_1} \varphi_r^n)- \mathcal{P}_{r-s_2}^H f({\mathbb{E}}^{s_2} B_r^H+ {\mathbb{E}}^{s_1} \varphi_{k_n(r)}^n) \\ & \qquad\qquad-\mathcal{P}_{r-s_2}^H f({\mathbb{E}}^{s_2} B_r^H + {\mathbb{E}}^{s_2} \varphi_r^n)+ \mathcal{P}_{r-s_3}^H f({\mathbb{E}}^{s_2} B_r^H+{\mathbb{E}}^{s_2} \varphi_{k_n(r)}^n)\mathrm{d} r \\ =&:I_1+I_2. \end{align}\] The two terms are treated in the exactly same manner, so we only detail \(I_1\). Similarly to 32 , we get \[\begin{align} \| I_1 \|_{L_\omega^p} \leqslant&C \| f\|_{{\mathcal{C}}^\alpha}\int_{s_4}^{s_5} (r-s_3)^{-(1-\alpha)H}\big\| {\mathbb{E}}^{s_1} | {\mathbb{E}}^{s_3} \varphi_r^n -{\mathbb{E}}^{s_3} \varphi_{k_n(r)}^n -{\mathbb{E}}^{s_1} \varphi_r^n + {\mathbb{E}}^{s_1} \varphi_{k_n(r)}^n | \big\|_{L_\omega^p} \nonumber\\ & \qquad+ (r-s_3)^{-(2-\alpha)H} \big\| | {\mathbb{E}}^{s_1} \varphi_r^n -{\mathbb{E}}^{s_1} \varphi_{k_n(r)}^n | \cdot {\mathbb{E}}^{s_1} | {\mathbb{E}}^{s_3} \varphi_{k_n(r)}^n-{\mathbb{E}}^{s_1} \varphi_{k_n(r)}^n | \big\|_{L_\omega^p} \mathrm{d} r.\label{eq:I95195bound952} \end{align}\tag{63}\] Similar to 34 , we have \[\label{eq:E1E361E1952} \| {\mathbb{E}}^{s_1}|{\mathbb{E}}^{s_3} \varphi_r^n-{\mathbb{E}}^{s_3} \varphi_{k_n(r)}^n-{\mathbb{E}}^{s_1} \varphi_r^n+ {\mathbb{E}}^{s_1} \varphi_{k_n(r)}^n | \|_{L^p_\omega} \leqslant\| {\mathbb{E}}^{s_1} | (\varphi_r^n-\varphi_{r}^n)-{\mathbb{E}}^{s_1} (\varphi_r^n-\varphi_{r)}^n) | \|_{L^p_\omega}.\tag{64}\] We note that \[\begin{align} \varphi_r^n-\varphi_{k_n(r)}^n &=\int_{k_n(r)}^r b(B^H_{k_n(t)}+\varphi_{k_n(t)}^n) \mathrm{d}t =(r-k_n(r)) b(B_{k_n(r)}^H+\varphi^n_{k_n(r)}) \in \mathcal{F}_{k_n(r)}. \end{align}\] When \(s_1 \in [k_n(r),r],\) we have \[\varphi_r^n-\varphi_{k_n(r)}^n-{\mathbb{E}}^{s_1}(\varphi_r^n-\varphi_{k_n(r)}^n)=0;\] when \(s_1 < k_n(r),\) by taking \(X=b(B_{k_n(r)}^H+\varphi^n_{k_n(r)})\) and \(Y=b({\mathbb{E}}^{s_1} B_{k_n(r)}^H+\varphi^n_{s_1}) \in \mathcal{F}_{s_1}\) in 8 , we obtain \[\begin{align} &{\mathbb{E}}^{s_1} | (\varphi_r^n-\varphi_{k_n(r)}^n)-{\mathbb{E}}^{s_1} (\varphi_r^n-\varphi_{k_n(r)}^n)| \\ =&(r-k_n(r)) {\mathbb{E}}^{s_1} |b(B_{k_n(r)}^H + \varphi_{k_n(r)}^n)-{\mathbb{E}}^{s_1} b(B_{k_n(r)}^H + \varphi_{k_n(r)}^n)| \\ \leqslant& 2 (r-k_n(r)) {\mathbb{E}}^{s_1} | b(B^H_{k_n(r)}+ \varphi_{k_n(r)}^n)-b({\mathbb{E}}^{s_1} B_{k_n(r)}^H + \varphi_{s_1}^n)|\\ \leqslant& C(r-k_n(r)) {\mathbb{E}}^{s_1} (|B_{k_n(r)}^H-{\mathbb{E}}^{s_1} B_{k_n(r)}^H|^\alpha + | \varphi_{k_n(r)}^n-\varphi_{s_1}^n|^\alpha). \end{align}\] Moreover, using 19 and 26 , we have \[\begin{align} &{\mathbb{E}}^{s_1} | (\varphi_r^n-\varphi_{k_n(r)}^n)-{\mathbb{E}}^{s_1} (\varphi_r^n-\varphi_{k_n(r)}^n)| \\ \leqslant& C(r-k_n(r)) (|k_n(r)-s_1|^{\alpha H} + |k_n(r)-s_1|^{\alpha}) \leqslant\frac{C}{n} \|b\|_{{\mathcal{C}}^\alpha} |r-s_1|^\alpha \end{align}\] where we used the fact \(H>1\) in the second inequality. Plugging it into 64 , we get \[\label{eq:cor4463-I1-1} \| {\mathbb{E}}^{s_1}|{\mathbb{E}}^{s_3} \varphi_r^n-{\mathbb{E}}^{s_3} \varphi_{k_n(r)}^n-{\mathbb{E}}^{s_1} \varphi_r^n+ {\mathbb{E}}^{s_1} \varphi_{k_n(r)}^n | \|_{L^p_\omega} \leqslant\frac{C}{n} |r-s_1|^\alpha.\tag{65}\]
Meanwhile 33 and 26 yield \[\begin{align} {\mathbb{E}}^{s_1} |{\mathbb{E}}^{s_3} \varphi_{r}^n-{\mathbb{E}}^{s_1} \varphi_{r}^n| \leqslant C |r-s_1|^{1+\alpha H},\tag{66}\\ \| {\mathbb{E}}^{s_1} (\varphi_r^n-\varphi_{k_n(r)}^n) \|_{L_\omega^p} \leqslant C \| \varphi_\cdot^n-\varphi_{k_n(\cdot)}^n \|_{C_p^0} \leqslant\frac{C}{n}.\tag{67} \end{align}\] Applying 65 , 66 and 67 into 63 gives us
\[\begin{align} \label{est:III1} \| I_1 \|_{L^p_\omega} &\leqslant C \frac{ \| f\|_{{\mathcal{C}}^\alpha}}{n} \int_{s_4}^{s_5} (r-s_3)^{-(1-\alpha)H} (r-s_1)^\alpha \mathrm{d} r \nonumber + (r-s_3)^{-(2-\alpha)H} (r-s_1)^{1+\alpha H}\mathrm{d}r \nonumber\\ & \leqslant C \frac{\|f\|_{{\mathcal{C}}^\alpha}}{n} (t-s)^{(1+\alpha-(1-\alpha)H) \wedge (2+(2 \alpha -2)H)}. \end{align}\tag{68}\]
The same bound holds on \(I_2\). Hence we get 44 . ◻
CL is supported by Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 563883019. KS is grateful to the financial supports by National Key R & D Program of China (No. 2022YFA1006300) and the financial supports of the NSFC (No. 12426205, No. 12271030).
We are grateful to Prof. Rongchan Zhu (Beijing Institute of Technology) for her fruitful advices. We also are grateful for valuable suggestion on the optimality result from an anonymous reviewer.