Euler-Maruyama scheme for SDE driven by Lévy process with Hölder drift

Yanfang Li and Guohuan Zhao Institute of Applied Mathematics, Academy of Mathematics and Systems Science, CAS, Beijing, 100190, China liyanfang@amss.ac.cn Institute of Applied Mathematics, Academy of Mathematics and Systems Science, CAS, Beijing, 100190, China gzhao@amss.ac.cn

Abstract.

This study focuses on approximating solutions to SDEs driven by Lévy processes with Hölder continuous drifts using the Euler-Maruyama scheme. We derive the $L^{p}$ -error for a broad range of driven noises, including all nondegenerate $\alpha$ -stable processes ( $0<\alpha<2$ ).

Research Guohuan is supported by the National Natural Science Foundation of China (No. 12288201); Research Yanfang is supported by the China Postdoctoral Science Foundation (No. 2022M723328)

Keywords: Stochastic differential equation, Lévy process, Euler-Maruyama scheme

AMS 2020 Mathematics Subject Classification: Primary 39A50; Secondary 41A25, 60J76

1. Introduction

Let $Z$ be a pure jump Lévy process whose characteristic exponent is given by

\psi(\xi)=\log{\mathbf{E}}\mathrm{e}^{i\xi\cdot Z_{1}}=\int_{{\mathbb{R}}^{d}}\left(\mathrm{e}^{i\xi\cdot z}-1-i\xi\cdot z{\mathbf{1}}_{B_{1}}(z)\right)\nu(\text{\rm{d}}z),

where $\nu$ is the intensity measure satisfying $\int_{{\mathbb{R}}^{d}}(1\wedge|z|^{2})\nu(\text{\rm{d}}z)<\infty$ . Consider the following stochastic differential equation (SDE) driven by $Z$ :

(1.1)

X_{t}=X_{0}+\int_{0}^{t}b(X_{s})\text{\rm{d}}s+Z_{t},

as well as its Euler-Maruyama scheme

(1.2)

X_{t}^{n}=X_{0}^{n}+\int_{0}^{t}b(X^{n}_{k_{n}(s)})\text{\rm{d}}s+Z_{t},

where $k_{n}(t):=[nt]/n$ .

The main purpose of this paper is to study the strong convergence rate of the Euler-Maruyama approximation for (1.1), where $Z$ belongs to a wide class of Lévy processes. The main result is formulated as follows:

Theorem 1.1.

Assume that there are constants $c_{0}>0$ , $\alpha\in(0,2)$ and $M>0$ such that

(1.3)

{\mathrm{Re}}(-\psi(\xi))\geqslant c_{0}|\xi|^{\alpha},\ \mbox{ for all }|\xi|\geqslant M,

and $b\in C^{\beta}$ with $\beta\in(1-\alpha/2,1)$ . Then for each $p>0$ , there is a constant C depending on $d,c_{0},M,\alpha,\beta,p,\|b\|_{\beta}$ such that

(1.4)

{\mathbf{E}}\sup_{t\in[0,1]}|X_{t}^{n}-X_{t}|^{p}\leqslant C\left[{\mathbf{E}}|X^{n}_{0}-X_{0}|^{p}+{\mathbf{E}}\left(1\wedge|Z_{1/n}|^{p\beta}\right)\right].

Remark 1.2.

One sufficient condition for $Z$ to satisfy (1.3) is

(1.5)

\int_{|z|\leqslant\rho}|\eta\cdot z|^{2}\nu(\text{\rm{d}}z)\geqslant c\rho^{2-\alpha},\ \forall\eta\in{\mathbb{S}}^{d-1},\rho\in(0,\rho_{0}],

where $c>0$ and $\rho_{0}>0$ are two positive constants. The reason is: using $1-\cos t\geqslant t^{2}/3\ (t\in[-1,1])$ , we have

	$\displaystyle\int_{{\mathbb{R}}^{d}}\left(1-\cos(\xi\cdot z)\right)\nu(\text{\rm{d}}z)\geqslant$	$\displaystyle c\int_{\|z\|\leqslant\|\xi\|^{-1}}\|z\cdot\xi\|^{2}\nu(\text{\rm{d}}z)$
	$\displaystyle\geqslant$	$\displaystyle c\|\xi\|^{2}\int_{\|z\|\leqslant\|\xi\|^{-1}}\|\hat{\xi}\cdot z\|^{2}\nu(\text{\rm{d}}z)\overset{\eqref{eq-nondege}}{\geqslant}c_{0}\|\xi\|^{\alpha},\quad\forall\,\|\xi\|\gg 1.$

The Euler-Maruyama approximation of stochastic differential equations (SDEs) is a well-established field of research in probability theory and numerical analysis, with a vast body of literature dedicated to it. A notable phenomenon in this area is the regularization of the noise for schemes with irregular drift. For instance, Gyöngy-Krylov [GK96] established the convergence (without an explicit rate) of the Euler-Maruyama scheme when the driven noise is the Brownian motion and the drift coefficient satisfies certain integrability conditions. Recently, researchers have imposed $\beta$ -Hölder type conditions on the modulus of continuity of drifts in [NT17], [BHY19], and [SYZ22] (with the latter two works discussing more general cases). Although the index $\beta$ can be arbitrarily small, the drawback is that the convergence rates obtained become increasingly worse as $\beta$ approaches zero. When the driven noise is an $\alpha$ -stable process with $\alpha\in(0,2)$ , and the drift coefficient is only $\beta$ -Hölder continuous with $\beta>1-\alpha/2$ , the strong well-posedness of (1.1) has been studied in [TTW74], [Pri12], [Pri15],[CSZ18], and [CZZ21]. However, only for $\alpha\in[1,2)$ , the rate of strong convergence for the Euler-Maruyama approximation of SDE (1.1) has been studied in [MPT17], [MX18], [HL18], and [KS19]. Notably, all of the convergence rates obtained in these works depend on the regularity of the drift coefficient, and they become increasingly worse as $\beta$ approaches $1-\alpha/2$ .

Our contribution is to relax some constraints on noise in previous studies. Specifically, we address a scenario where the intensity measure $\nu$ is singular with respect to the Lebesgue measure and the parameter $\alpha$ lies in the whole range $(0,2)$ . Our approach is quite straightforward. In section 2,we consider the following resolvent equation that corresponds to (1.1)

(1.6)

\lambda u-Lu-b\cdot\nabla u=f,

where $L$ is the infinitesimal generator of $Z$ , i.e.

Lu(x)=\int_{{\mathbb{R}}^{d}}\left(f(x+z)-f(x)-\nabla f(x)\cdot z{\mathbf{1}}_{B_{1}}\right)~{}\nu(\text{\rm{d}}z).

Using the similar line of proof from the second named author’s previous work [Zha21, Theorem 1.1] (see also [CZZ21]), we arrive at the primary auxiliary analytic result, Theorem 2.3, which establishes a good regularity estimate for solutions to (1.6). Then in section 3, by re-expressing the drift term in equations (1.1) and (1.2) in a form that facilitates comparison between the two, we prove our main result.

We close this section by mentioning the remarkable contributions of recent works, such as [DG20], [BDG21] and [BJ22], which have shown that an almost $1/2$ rate of convergence holds for all Hölder (or Dini) continuous coefficients, when the driven noises are Brownian motions. These results are further supported by related works such as [LS17] and [MGY20]. However, to the best of our knowledge, there is currently no literature that investigates whether the convergence rate is insensitive to the regularity of $b$ when $Z$ is modeled by an $\alpha$ -stable process. This issue is beyond the scope of this short note, so we will investigate it in our future work.

2. Auxiliary Results

In this section, we formulate some results that will be used in the proof of Theorem 1.1. Before that let us introduce some notions and recall very basic facts from Littlewood-Paley theory. Let ${\mathscr{S}}({\mathbb{R}}^{d})$ be the Schwartz space of all rapidly decreasing functions, and ${\mathscr{S}}^{\prime}({\mathbb{R}}^{d})$ the dual space of ${\mathscr{S}}({\mathbb{R}}^{d})$ called Schwartz generalized function (or tempered distribution) space. Denote the Fourier transform of $f\in{\mathscr{S}}^{\prime}({\mathbb{R}}^{d})$ by ${\mathcal{F}}f$ or $\hat{f}$ .

Let $\chi:{\mathbb{R}}^{d}\to[0,1]$ be a smooth radial function so that $\chi|_{B_{3/4}}=1$ and $\chi|_{B_{1}^{c}}=0$ . Define

\varphi(\xi):=\chi(\xi)-\chi(2\xi).

The dyadic block operator $\Delta_{j}$ is defined by

\Delta_{j}f:=\left\{\begin{array}[]{ll}{\mathcal{F}}^{-1}(\chi(2\cdot){\mathcal{F}}f),&j=-1,\\ {\mathcal{F}}^{-1}(\varphi(2^{-j}\cdot){\mathcal{F}}f),&j\geqslant 0.\end{array}\right.

For $s\in{\mathbb{R}}$ and $p\in[1,\infty]$ , the Besov space $B^{s}_{p,p}$ is defined as the set of all $f\in{\mathscr{S}}^{\prime}({\mathbb{R}}^{d})$ with

\|f\|_{B^{s}_{p,p}}:={\mathbf{1}}_{\{p<\infty\}}\left(\sum_{j\geqslant-1}2^{jsp}\|\Delta_{j}f\|_{p}^{p}\right)^{1/p}+{\mathbf{1}}_{\{p=\infty\}}\left(\sup_{j\geqslant-1}2^{js}\|\Delta_{j}f\|_{p}\right)<\infty.

For each $s>0$ with $s\notin{\mathbb{N}}$ , and $p\in[1,\infty)$ , the Hölder space and Sobolev-Slobodeckij space are defined by

\|f\|_{C^{s}}=\sum_{0\leqslant k\leqslant[s]}\|\nabla^{k}f\|_{L^{\infty}}+\sup_{x\neq y}\frac{|\nabla^{[s]}f(x)-\nabla^{[s]}f(y)|}{|x-y|^{s-[s]}}

and

\|f\|_{W^{s}_{p}}:=\sum_{0\leqslant k\leqslant[s]}\|\nabla^{k}f\|_{L^{p}}+\left(\int\!\!\!\int_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\frac{|\nabla^{[s]}f(x)-\nabla^{[s]}f(y)|}{|x-y|^{d+(s-[s])p}}\right)^{1/p},

respectively. We have the following relations for the above functional spaces:

B^{s}_{p,p}=W^{s}_{p},\quad B^{s}_{\infty,\infty}=C^{s}

(see for instance [Tri92]).

In the following, we present some analytical results that are required to prove our main theorem. Firstly, we introduce a novel form of Bernstein’s inequality, as presented in [CMZ07].

Lemma 2.1.

For any $2\leqslant p<\infty$ , $j\geqslant 0$ and $\alpha\in(0,2)$ , there is a constant $c>0$ such that for all $f\in{\mathscr{S}}^{\prime}({\mathbb{R}}^{d})$ ,

(2.1)

\displaystyle\int_{{\mathbb{R}}^{d}}\Big{|}(-\Delta)^{\alpha/4}|\Delta_{j}f|^{p/2}\Big{|}^{2}\geqslant c2^{\alpha j}\|\Delta_{j}f\|_{p}^{p}.

Secondly, following [CZZ21], we have the following crucial lemma.

Lemma 2.2.

Suppose ${\mathrm{Re}}(-\psi(\xi))\geqslant c_{0}|\xi|^{\alpha}$ for some $c_{0}>0$ and $|\xi|\geqslant M>0$ . Then for any $p>2$ , there are constants $c_{p}=c(c_{0},p)>0$ and $j_{0}=j_{0}(c_{0},M)$ such that for all $j\geqslant j_{0}$ , it holds that

(2.2)

\displaystyle\int_{{\mathbb{R}}^{d}}|\Delta_{j}f|^{p-2}(\Delta_{j}f)L\Delta_{j}f\leqslant-c_{p}2^{\alpha j}\|\Delta_{j}f\|_{p}^{p},

and for all $j=-1,0,1,\cdots$ ,

(2.3)

\int_{{\mathbb{R}}^{d}}|\Delta_{j}f|^{p-2}(\Delta_{j}f)L\Delta_{j}f\leqslant 0.

Proof.

For $p\geqslant 2$ , by the elementary inequality $|r|^{p/2}-1\geqslant\frac{p}{2}(r-1)$ for $r\in{\mathbb{R}}$ , we have

|a|^{p/2}-|b|^{p/2}\geqslant\tfrac{p}{2}(a-b)b|b|^{p/2-2},\ \ a,b\in{\mathbb{R}}.

Letting $g$ be a smooth function, by definition we have

	$\displaystyle L\|g\|^{p/2}(x)$	$\displaystyle=\int_{{\mathbb{R}}^{d}}\Big{(}\|g(x+z)\|^{p/2}-\|g(x)\|^{p/2}-{\mathbf{1}}_{B_{1}}(z)z\cdot\nabla\|g(x)\|^{p/2}\Big{)}\nu(\text{\rm{d}}z)$
		$\displaystyle\geqslant\frac{p}{2}\|g(x)\|^{p/2-2}g(x)\int_{{\mathbb{R}}^{d}}\Big{(}g(x+z)-g(x)-{\mathbf{1}}_{B_{1}}(z)z\cdot\nabla g(x)\Big{)}\nu(\text{\rm{d}}z)$
		$\displaystyle=\frac{p}{2}\|g(x)\|^{p/2-2}g(x)Lg(x).$

Multiplying both sides by $|g|^{p/2}$ and then integrating in $x$ over ${\mathbb{R}}^{d}$ , by Plancherel’s formula, we obtain

	$\displaystyle\int_{{\mathbb{R}}^{d}}\|g\|^{p-2}g\,Lg$	$\displaystyle\leqslant\frac{2}{p}\int_{{\mathbb{R}}^{d}}\|g\|^{p/2}L\|g\|^{p/2}=\frac{2}{p}\int_{{\mathbb{R}}^{d}}\|{\mathcal{F}}(\|g\|^{p/2})(\xi)\|^{2}\psi(\xi)\text{\rm{d}}\xi$
		$\displaystyle=\frac{2}{p}\int_{{\mathbb{R}}^{d}}\|{\mathcal{F}}(\|g\|^{p/2})(\xi)\|^{2}\mathrm{Re}(\psi(\xi))\text{\rm{d}}\xi\leqslant 0,$

which implies (2.3). Moreover, by our assumption on $\psi$ , we have $\mathrm{Re}(-\psi(\xi))\geqslant c_{0}|\xi|^{\alpha}-M^{\alpha}$ (for all $\xi\in{\mathbb{R}}^{d}$ ). Thus,

	$\displaystyle\int_{{\mathbb{R}}^{d}}\|g\|^{p-2}g\,Lg$	$\displaystyle\leqslant-\frac{2c_{0}}{p}\int_{{\mathbb{R}}^{d}}\|{\mathcal{F}}(\|g\|^{p/2})(\xi)\|^{2}(\|\xi\|^{\alpha}-M^{\alpha})\text{\rm{d}}\xi$
		$\displaystyle\leqslant-c(c_{0},d,p)\int_{{\mathbb{R}}^{d}}\|(-\Delta)^{\alpha/4}\|g\|^{p/2}\|^{2}\text{\rm{d}}x+{c_{0}M^{\alpha}}\\|g\\|_{p}^{p}.$

This in turn gives (2.2) by (2.1) (taking $g=\Delta_{j}f$ ). Inequality (2.3) is trivial, so we omit its proof here. ∎

The following result is a refined version of [Zha21, Theorem 1.1] (with $\sigma={\mathbb{I}}$ therein).

Theorem 2.3.

Under the same assumptions of Theorem 1.1, there exists a constant $\lambda_{0}=\lambda_{0}(d,\alpha,c_{0},M,\beta,\|b\|_{\beta})$ such that for any $\lambda\geqslant\lambda_{0}$ , equation (1.6) has a unique solution $u\in\bigcap_{\gamma<\beta}C^{\alpha+\gamma}$ . Moreover, it holds that

(2.4)

\lambda\|u\|_{C^{\gamma}}+\|u\|_{C^{\alpha+\gamma}}\leqslant C\|f\|_{C^{\beta}},

where $C$ only depends on $d,\alpha,c_{0},M,\beta,\gamma$ and $\|b\|_{C^{\beta}}$ .

Proof.

For all $p\in[2,\infty)$ , replacing [Zha21, Lemma 3.1] by our Lemma 2.2 and following the proof for [Zha21, Theorem 3.8], one can see that

(2.5)

\sup_{z\in{\mathbb{R}}^{d}}\left(\lambda\|u\chi(\cdot+z)\|_{W^{\beta}_{p}}+\|u\chi(\cdot+z)\|_{W^{\alpha+\beta}_{p}}\right)\leqslant C\sup_{z\in{\mathbb{R}}^{d}}\|f\chi(\cdot+z)\|_{W^{\beta}_{p}}.

Due to [Zha21, Lemma 2.6], for any $s>d/p$ and $\varepsilon>0$ , it holds that

\|u\|_{C^{s-d/p}}\leqslant C\sup_{z\in{\mathbb{R}}^{d}}\|u\chi(\cdot+z)\|_{W^{s}_{p}}\leqslant C\|u\|_{C^{s+\varepsilon}}.

Thus, for any $0<\gamma<\theta<\beta$ and $p=d/(\theta-\gamma)$ , we obtain

	$\displaystyle\lambda\\|u\\|_{C^{\gamma}}+\\|u\\|_{C^{\alpha+\gamma}}\leqslant$	$\displaystyle\sup_{z\in{\mathbb{R}}^{d}}\left(\lambda\\|u\chi(\cdot+z)\\|_{W^{\theta}_{p}}+\\|u\chi(\cdot+z)\\|_{W^{\alpha+\theta}_{p}}\right)$
	$\displaystyle\overset{\eqref{eq-WC}}{\leqslant}$	$\displaystyle C\sup_{z\in{\mathbb{R}}^{d}}\\|f\chi(\cdot+z)\\|_{W^{\theta}_{p}}\leqslant C\\|f\\|_{C^{\beta}}.$

∎

3. Proof of main result

Proof of Theorem 1.1.

Fix $\gamma\in(1-\frac{\alpha}{2},\beta)$ . Based on our assumptions and Theorem 2.3, we can conclude that equation (1.6) with $f=b$ has a unique solution $u\in C^{\alpha+\gamma}$ . Furthermore, if $\theta:=\frac{\alpha+\gamma-1}{\alpha}>0$ and $\lambda$ is large enough, by interpolation, the following inequality holds:

(3.1)

\|u\|_{C^{1}}\leqslant\|u\|_{C^{\gamma}}^{\theta}\|u\|_{C^{\alpha+\gamma}}^{1-\theta}\leqslant C\lambda^{-\theta}\leqslant\frac{1}{4}.

Let $N(\text{\rm{d}}r,\text{\rm{d}}z)$ be the Poisson random measure associated with $Z$ , whose intensity measure is given by $\text{\rm{d}}r\,\nu(\text{\rm{d}}z)$ . Then

Z_{t}=\int_{0}^{t}\!\!\!\int_{|z|<1}z\widetilde{N}(\text{\rm{d}}r,\text{\rm{d}}z)+\int_{0}^{t}\!\!\!\int_{|z|\geqslant 1}z{N}(\text{\rm{d}}r,\text{\rm{d}}z),

where $\widetilde{N}(\text{\rm{d}}r,\text{\rm{d}}z)=N(\text{\rm{d}}r,\text{\rm{d}}z)-\text{\rm{d}}r\nu(\text{\rm{d}}z)$ . Thanks to the generalized Itô’s formula (see [Pri12]), we have

		$\displaystyle u(X^{n}_{t})-u(X^{n}_{s})$
	$\displaystyle=$	$\displaystyle\int_{s}^{t}\Big{(}Lu(X^{n}_{r})+b(X^{n}_{k_{n}(r)})\cdot\nabla u(X^{n}_{r})\Big{)}\,\text{\rm{d}}r+\int_{s}^{t}\!\!\!\int_{{\mathbb{R}}^{d}}\Big{(}u(X^{n}_{r-}+z)-u(X^{n}_{r-})\Big{)}\widetilde{N}(\text{\rm{d}}r,\text{\rm{d}}z)$
	$\displaystyle=$	$\displaystyle\int_{s}^{t}(\lambda u-b)(X^{n}_{r})\text{\rm{d}}r+\int_{s}^{t}\Big{(}b(X^{n}_{k_{n}(r)}-b(X^{n}_{r}))\Big{)}\cdot\nabla u(X^{n}_{r})\text{\rm{d}}r$
		$\displaystyle+\int_{s}^{t}\!\!\!\int_{{\mathbb{R}}^{d}}\Big{(}u(X^{n}_{r-}+z)-u(X^{n}_{r-})\Big{)}\widetilde{N}(\text{\rm{d}}r,\text{\rm{d}}z),$

which implies

(3.2)		$\displaystyle\int_{s}^{t}b(X^{n}_{r})\text{\rm{d}}r=$	$\displaystyle u(X^{n}_{s})-u(X^{n}_{t})+\lambda\int_{s}^{t}u(X^{n}_{r})\text{\rm{d}}r+\int_{s}^{t}\Big{(}b(X^{n}_{k_{n}(r)}-b(X^{n}_{r}))\Big{)}\cdot\nabla u(X^{n}_{r})\text{\rm{d}}r$
(3.2)			$\displaystyle+\int_{s}^{t}\!\!\!\int_{{\mathbb{R}}^{d}}\Big{(}u(X^{n}_{r-}+z)-u(X^{n}_{r-})\Big{)}\widetilde{N}(\text{\rm{d}}r,\text{\rm{d}}z).$

Similarly,

(3.3)		$\displaystyle\int_{s}^{t}b(X_{r})\text{\rm{d}}r=$	$\displaystyle u(X_{s})-u(X_{t})+\lambda\int_{s}^{t}u(X_{r})\text{\rm{d}}r$
(3.3)			$\displaystyle+\int_{s}^{t}\!\!\!\int_{{\mathbb{R}}^{d}}\Big{(}u(X_{r-}+z)-u(X_{r-})\Big{)}\widetilde{N}(\text{\rm{d}}r,\text{\rm{d}}z).$

By definition,

\displaystyle X^{n}_{t}-X_{t}=

\displaystyle X_{0}^{n}-X_{0}+\int_{0}^{t}\Big{(}b(X^{n}_{k_{n}(r)})-b(X^{n}_{r})\Big{)}\text{\rm{d}}r+\int_{0}^{t}\Big{(}b(X^{n}_{r})-b(X_{r})\Big{)}\text{\rm{d}}r

Plugging (3.2) and (3.3) in to the above equation, we obtain

	$\displaystyle X^{n}_{t}-X_{t}=$	$\displaystyle X_{0}^{n}-X_{0}+\int_{0}^{t}\Big{(}b(X^{n}_{k_{n}(r)})-b(X^{n}_{r})\Big{)}\text{\rm{d}}r+\Big{(}u(X^{n}_{0})-u(X_{0})\Big{)}+\Big{(}u(X_{t})-u(X^{n}_{t})\Big{)}$
		$\displaystyle+\lambda\int_{0}^{t}\Big{(}u(X^{n}_{r})-u(X_{r})\Big{)}\text{\rm{d}}r+\int_{0}^{t}\Big{(}b(X^{n}_{k_{n}(r)}-b(X^{n}_{r}))\Big{)}\cdot\nabla u(X^{n}_{r})\text{\rm{d}}r$
		$\displaystyle+\int_{0}^{t}\!\!\!\int_{{\mathbb{R}}^{d}}\left[\Big{(}u(X^{n}_{r-}+z)-u(X^{n}_{r-})\Big{)}-\Big{(}u(X_{r-}+z)-u(X_{r-})\Big{)}\right]\widetilde{N}(\text{\rm{d}}r,\text{\rm{d}}z)$
	$\displaystyle=$	$\displaystyle:\sum_{i=1}^{7}I_{i}.$

For $I_{2}$ and $I_{6}$ , by our assumption on $b$ and (3.1), we have

{\mathbf{E}}|I_{2}|^{p}\lesssim\int_{0}^{t}{\mathbf{E}}\left(1\wedge|X^{n}_{k_{n}(r)}-X^{n}_{r}|^{\beta}\right)^{p},\quad{\mathbf{E}}|I_{6}|^{p}\lesssim\int_{0}^{t}{\mathbf{E}}\left(1\wedge|X^{n}_{k_{n}(r)}-X^{n}_{r}|^{\beta}\right)^{p}.

For $I_{3}$ , $I_{4}$ and $I_{5}$ , again using (3.1), one sees that

|I_{3}|\leqslant\frac{1}{4}|X_{0}^{n}-X_{0}|,\quad|I_{4}|\leqslant\frac{1}{4}|X_{t}^{n}-X_{t}|,\quad I_{5}\leqslant C\lambda^{1-\theta}\int_{0}^{t}|X^{n}_{r}-X_{r}|\text{\rm{d}}r.

For $I_{7}$ , by BDG inequatlity and the basic fact that $|f(x+z)-f(x)-f(y+z)+f(y)|\lesssim\|f\|_{C^{\alpha+\gamma}}|x-y|(1\wedge|z|^{\alpha+\gamma-1})$ , we get

	$\displaystyle{\mathbf{E}}\|I_{7}\|^{p}\lesssim$	$\displaystyle{\mathbf{E}}\left(\int_{0}^{t}\!\!\!\int_{{\mathbb{R}}^{d}}\|X^{n}_{r-}-X_{r-}\|^{2}(1\wedge\|z\|)^{2\alpha+2\gamma-2}\nu(\text{\rm{d}}z)\,\text{\rm{d}}r)\right)^{p/2}$
	$\displaystyle\lesssim$	$\displaystyle{\mathbf{E}}\left(\int_{0}^{t}\|X^{n}_{r}-X_{r}\|^{2}\text{\rm{d}}r\right)^{p/2}.$

Combining all the estimates above, we arrive

	$\displaystyle{\mathbf{E}}\sup_{t\in[0,T_{0}]}\|X^{n}_{t}-X_{t}\|^{p}\lesssim$	$\displaystyle{\mathbf{E}}\|X^{n}_{0}-X_{0}\|^{p}+(T_{0}^{p}+T_{0}^{p/2}){\mathbf{E}}\sup_{t\in[0,T_{0}]}\|X^{n}_{t}-X_{t}\|^{p}$
		$\displaystyle+\int_{0}^{T_{0}}{\mathbf{E}}(1\wedge\|X^{n}_{k_{n}(t)}-X^{n}_{t}\|^{\beta p})\text{\rm{d}}t.$

Therefore, for $T_{0}\in(0,1]$ sufficiently small, we have

	$\displaystyle{\mathbf{E}}\sup_{t\in[0,T_{0}]}\|X^{n}_{t}-X_{t}\|^{p}\lesssim$	$\displaystyle{\mathbf{E}}\|X^{n}_{0}-X_{0}\|^{p}+\int_{0}^{T_{0}}{\mathbf{E}}\left(1\wedge\|Z_{k_{n}(t)}-Z_{t}\|^{\beta p}\right)\text{\rm{d}}t$
	$\displaystyle\lesssim$	$\displaystyle{\mathbf{E}}\|X^{n}_{0}-X_{0}\|^{p}+{\mathbf{E}}(1\wedge\|Z_{1/n}\|^{\beta p}).$

Iterating this procedure finite times we reach the full time horizon $[0,1]$ . ∎

Corollary 3.1.

If $Z$ is a nondegenerate $\alpha$ -stable process, then

\displaystyle{\mathbf{E}}\sup_{t\in[0,1]}|X_{t}^{n}-X_{t}|^{p}\leqslant C{\mathbf{E}}|X^{n}_{0}-X_{0}|^{p}+C\left\{\begin{aligned} &n^{-p\beta/\alpha},&&\mbox{ if }0<p<\alpha/\beta,\\ &n^{-1}\log n,&&\mbox{ if }p=\alpha/\beta,\\ &n^{-1},&&\mbox{ if }p>\alpha/\beta.\end{aligned}\right.

Proof.

Since $Z_{t}\overset{d}{=}t^{1/\alpha}Z_{1}$ , we have

	$\displaystyle{\mathbf{E}}\left(1\wedge\|Z_{1/n}\|^{p}\right)=$	$\displaystyle{\mathbf{E}}\left(1\wedge n^{-\frac{p}{\alpha}}\|Z_{1}\|^{p}\right)$
	$\displaystyle=$	$\displaystyle{\mathbf{P}}(\|Z_{1}\|>n^{\frac{1}{\alpha}})+n^{-\frac{p}{\alpha}}{\mathbf{E}}\left(\|Z_{1}\|^{p}{\mathbf{1}}_{\{\|Z_{1}\|\leqslant n^{1/\alpha}\}}\right)$
	$\displaystyle\lesssim$	$\displaystyle n^{-1}+n^{-\frac{p}{\alpha}}\int_{0}^{n^{1/\alpha}}t^{p-1}{\mathbf{P}}(\|Z_{1}\|>t)\text{\rm{d}}t$
	$\displaystyle\lesssim$	$\displaystyle n^{-1}+n^{-\frac{p}{\alpha}}+n^{-\frac{p}{\alpha}}\int_{1}^{n^{1/\alpha}}t^{p-\alpha-1}\text{\rm{d}}t$
	$\displaystyle\lesssim$	$\displaystyle\left\{\begin{aligned} &n^{-p/\alpha},&&\mbox{ if }0<p<\alpha,\\ &n^{-1}\log n,&&\mbox{ if }p=\alpha,\\ &n^{-1},&&\mbox{ if }p>\alpha.\end{aligned}\right.$

So we complete our proof. ∎

References

[BDG21] Oleg Butkovsky, Konstantinos Dareiotis, and Máté Gerencsér. Approximation of SDEs: a stochastic sewing approach. Probab. Theory Related Fields, 181(4):975–1034, 2021.
[BHY19] Jianhai Bao, Xing Huang, and Chenggui Yuan. Convergence rate of Euler-Maruyama scheme for SDEs with Hölder-Dini continuous drifts. J. Theoret. Probab., 32(2):848–871, 2019.
[BJ22] Oumaima Bencheikh and Benjamin Jourdain. Convergence in total variation of the Euler-Maruyama scheme applied to diffusion processes with measurable drift coefficient and additive noise. SIAM J. Numer. Anal., 60(4):1701–1740, 2022.
[CMZ07] Qionglei Chen, Changxing Miao, and Zhifei Zhang. A new Bernstein’s inequality and the 2d dissipative quasi-geostrophic equation. Communications in mathematical physics, 271(3):821–838, 2007.
[CSZ18] Zhen-Qing Chen, Renming Song, and Xicheng Zhang. Stochastic flows for Lévy processes with Hölder drifts. Revista Matemática Iberoamericana, 34(4), 2018.
[CZZ21] Zhen-Qing Chen, Xicheng Zhang, and Guohuan Zhao. Supercritical SDEs driven by multiplicative stable-like Lévy processes. Transactions of the American Mathematical Society, 374(11):7621–7655, 2021.
[DG20] Konstantinos Dareiotis and Máté Gerencsér. On the regularisation of the noise for the Euler-Maruyama scheme with irregular drift. Electron. J. Probab., 25:Paper No. 82, 18, 2020.
[GK96] István Gyöngy and Nicolai Krylov. Existence of strong solutions for Itô’s stochastic equations via approximations. Probab. Theory Related Fields, 105(2):143–158, 1996.
[HL18] Xing Huang and Zhong-Wei Liao. The Euler-Maruyama method for S(F)DEs with Hölder drift and $\alpha$ -stable noise. Stoch. Anal. Appl., 36(1):28–39, 2018.
[KS19] Franziska Kühn and René L. Schilling. Strong convergence of the Euler-Maruyama approximation for a class of Lévy-driven SDEs. Stochastic Process. Appl., 129(8):2654–2680, 2019.
[LS17] Gunther Leobacher and Michaela Szölgyenyi. A strong order 1/2 method for multidimensional SDEs with discontinuous drift. Ann. Appl. Probab., 27(4):2383–2418, 2017.
[MGY20] Thomas Müller-Gronbach and Larisa Yaroslavtseva. On the performance of the Euler-Maruyama scheme for SDEs with discontinuous drift coefficient. Ann. Inst. Henri Poincaré Probab. Stat., 56(2):1162–1178, 2020.
[MPT17] Olivier Menoukeu Pamen and Dai Taguchi. Strong rate of convergence for the Euler-Maruyama approximation of SDEs with Hölder continuous drift coefficient. Stochastic Process. Appl., 127(8):2542–2559, 2017.
[MX18] R. Mikulevičius and Fanhui Xu. On the rate of convergence of strong Euler approximation for SDEs driven by Lévy processes. Stochastics, 90(4):569–604, 2018.
[NT17] Hoang-Long Ngo and Dai Taguchi. Strong convergence for the Euler-Maruyama approximation of stochastic differential equations with discontinuous coefficients. Statist. Probab. Lett., 125:55–63, 2017.
[Pri12] Enrico Priola. Pathwise uniqueness for singular SDEs driven by stable processes. Osaka Journal of Mathematics, 49(2):421–447, 2012.
[Pri15] Enrico Priola. Stochastic flow for SDEs with jumps and irregular drift term. Banach Center Publications, 105(1):421–44193–210, 2015.
[SYZ22] Yongqiang Suo, Chenggui Yuan, and Shao-Qin Zhang. Weak convergence of Euler scheme for SDEs with low regular drift. Numer. Algorithms, 90(2):731–747, 2022.
[Tri92] Hans. Triebel. Theory of function spaces II, volume 84. Birkhäuser Basel, 1992.
[TTW74] Hiroshi Tanaka, Masaaki Tsuchiya, and Shinzo Watanabe. Perturbation of drift-type for Lévy processes. Journal of Mathematics of Kyoto University, 14(1):73–92, 1974.
[Zha21] Guohuan Zhao. Regularity properties of jump diffusions with irregular coefficients. Journal of Mathematical Analysis and Applications, 502(1):125220, 2021.

	$\displaystyle L\|g\|^{p/2}(x)$	$\displaystyle=\int_{{\mathbb{R}}^{d}}\Big{(}\|g(x+z)\|^{p/2}-\|g(x)\|^{p/2}-{\mathbf{1}}_{B_{1}}(z)z\cdot\nabla\|g(x)\|^{p/2}\Big{)}\nu(\text{\rm{d}}z)$
		$\displaystyle\geqslant\frac{p}{2}\|g(x)\|^{p/2-2}g(x)\int_{{\mathbb{R}}^{d}}\Big{(}g(x+z)-g(x)-{\mathbf{1}}_{B_{1}}(z)z\cdot\nabla g(x)\Big{)}\nu(\text{\rm{d}}z)$
		$\displaystyle=\frac{p}{2}\|g(x)\|^{p/2-2}g(x)Lg(x).$

	$\displaystyle\lambda\\|u\\|_{C^{\gamma}}+\\|u\\|_{C^{\alpha+\gamma}}\leqslant$	$\displaystyle\sup_{z\in{\mathbb{R}}^{d}}\left(\lambda\\|u\chi(\cdot+z)\\|_{W^{\theta}_{p}}+\\|u\chi(\cdot+z)\\|_{W^{\alpha+\theta}_{p}}\right)$
	$\displaystyle\overset{\eqref{eq-WC}}{\leqslant}$	$\displaystyle C\sup_{z\in{\mathbb{R}}^{d}}\\|f\chi(\cdot+z)\\|_{W^{\theta}_{p}}\leqslant C\\|f\\|_{C^{\beta}}.$

	$\displaystyle{\mathbf{E}}\|I_{7}\|^{p}\lesssim$	$\displaystyle{\mathbf{E}}\left(\int_{0}^{t}\!\!\!\int_{{\mathbb{R}}^{d}}\|X^{n}_{r-}-X_{r-}\|^{2}(1\wedge\|z\|)^{2\alpha+2\gamma-2}\nu(\text{\rm{d}}z)\,\text{\rm{d}}r)\right)^{p/2}$
	$\displaystyle\lesssim$	$\displaystyle{\mathbf{E}}\left(\int_{0}^{t}\|X^{n}_{r}-X_{r}\|^{2}\text{\rm{d}}r\right)^{p/2}.$

	$\displaystyle{\mathbf{E}}\sup_{t\in[0,T_{0}]}\|X^{n}_{t}-X_{t}\|^{p}\lesssim$	$\displaystyle{\mathbf{E}}\|X^{n}_{0}-X_{0}\|^{p}+\int_{0}^{T_{0}}{\mathbf{E}}\left(1\wedge\|Z_{k_{n}(t)}-Z_{t}\|^{\beta p}\right)\text{\rm{d}}t$
	$\displaystyle\lesssim$	$\displaystyle{\mathbf{E}}\|X^{n}_{0}-X_{0}\|^{p}+{\mathbf{E}}(1\wedge\|Z_{1/n}\|^{\beta p}).$