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Strong uniform Wong--Zakai approximations of Lévy-driven Marcus SDEs

Ilya Pavlyukevich, Sooppawat Thipyarat

TL;DR

This paper analyzes strong Wong–Zakai type approximations for Lévy-driven Marcus SDEs, establishing both a global strong convergence rate of $1/2$ and a locally uniform rate of $(1-\varepsilon)/(4d)$ for the non-linear polygonal approximation scheme. The authors construct a Wong–Zakai scheme $X^h$ using polygonal approximations of the Brownian motion and a pure-jump Lévy process, and they lever the Marcus flow $\boldsymbol{\varphi}^z$ and the non-autonomous map $\boldsymbol{\ Ψ}$ to derive sharp error bounds. Under a comprehensive set of smoothness and integrability assumptions, they prove existence, uniqueness, and moment bounds for the Marcus SDE, then prove strong convergence of the scheme with explicit rates both globally (in initial data $x$) and locally (uniform on bounded sets in $x$). The locally uniform result is obtained via Kunita’s random-field framework, revealing how the convergence rate degrades with dimension as $h^{(1-\varepsilon)/(4d)}$; these results provide rigorous, dimension-aware guarantees for simulating Lévy-driven Marcus SDEs. This advances the numerical analysis of jump-diffusions and supports reliable simulations in applications requiring strong pathwise accuracy.

Abstract

For a solution $X$ of a Lévy-driven $d$-dimensional Marcus (canonical) stochastic differential equation, we show that the Wong--Zakai type approximation scheme $X^h$ has a strong convergence of order $\frac12$: for each $T\in [0,\infty)$ and all $x\in\mathbb R^d$ we have $$ \mathbf E \sup_{kh\leq T}|X_{kh}(x)-X^h_{kh}(x)|\leq C h^{\frac{1}{2}}(1+|x|),\quad h\to 0. $$ We also determine the rate of the locally uniform strong convergence: for each $N\in(0,\infty)$ and $\varepsilon\in (0,1)$ we have $$ \mathbf E\sup_{|x|\leq N}\sup_{kh\leq T}|X_{kh}(x)-X^h_{kh}(x)|\leq C h^{\frac{1-\varepsilon}{4d}},\quad h\to 0. $$

Strong uniform Wong--Zakai approximations of Lévy-driven Marcus SDEs

TL;DR

This paper analyzes strong Wong–Zakai type approximations for Lévy-driven Marcus SDEs, establishing both a global strong convergence rate of and a locally uniform rate of for the non-linear polygonal approximation scheme. The authors construct a Wong–Zakai scheme using polygonal approximations of the Brownian motion and a pure-jump Lévy process, and they lever the Marcus flow and the non-autonomous map to derive sharp error bounds. Under a comprehensive set of smoothness and integrability assumptions, they prove existence, uniqueness, and moment bounds for the Marcus SDE, then prove strong convergence of the scheme with explicit rates both globally (in initial data ) and locally (uniform on bounded sets in ). The locally uniform result is obtained via Kunita’s random-field framework, revealing how the convergence rate degrades with dimension as ; these results provide rigorous, dimension-aware guarantees for simulating Lévy-driven Marcus SDEs. This advances the numerical analysis of jump-diffusions and supports reliable simulations in applications requiring strong pathwise accuracy.

Abstract

For a solution of a Lévy-driven -dimensional Marcus (canonical) stochastic differential equation, we show that the Wong--Zakai type approximation scheme has a strong convergence of order : for each and all we have We also determine the rate of the locally uniform strong convergence: for each and we have

Paper Structure

This paper contains 8 sections, 13 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Setting and the main result
  3. Properties of the mappings $\varphi^z(\cdot)$ and $\Psi(\cdot)$
  4. Proof of Lemma \ref{['l:bounded']}
  5. Proof of Theorem \ref{['t:existenceSDE']}
  6. Proof of Theorem \ref{['t:existenceWZ']}
  7. Proof of Theorem \ref{['thm:Xx-Xhx']}
  8. Proof of Theorem $\ref{['t:main']}$

Key Result

Theorem 1.1

Let $a\in C^2_b(\mathbb{R}^d,\mathbb{R}^d)$ and $b\in C^2_b(\mathbb{R}^d,\mathbb{R}^{d\times m})$. Then for every $T\in[0,\infty)$ we have Moreover, for every $T\in[0,\infty)$ and $N\in(0,\infty)$ and $p\in[2,\infty)$ we have

Theorems & Definitions (17)

  • Theorem 1.1: IW89, Chapter VI, Theorem 7.2, Remark 7.2 and Theorem 7.3
  • Theorem 1.2: GyongyM-04, Remark 2.6 and Theorem 2.8
  • Theorem 1.3: Theorem 2.4 in BrzFla-95
  • Theorem 1.4: Theorem 2.3.2 in shmatkov2006rate
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Lemma 2.5
  • Theorem 2.6
  • ...and 7 more