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

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 $\frac{1}{2}$: for each $T\in [0,\infty )$ and all $x\in {\mathbb{R}}^d$ we have $$\mathbf{E}\underset{kh\le T}{\sup }|X_{kh}(x)-X_{kh}^h(x)|\le C{h}^{\frac{1}{2}}(1+|x|),h\to 0.$$ We also determine the rate of the locally uniform strong convergence: for each $N\in (0,\infty )$ and $\epsilon \in (0,1)$ we have $$\mathbf{E}\underset{|x|\le N}{\sup }\underset{kh\le T}{\sup }|X_{kh}(x)-X_{kh}^h(x)|\le C{h}^{\frac{1-\epsilon }{4d}},h\to 0.$$

18 pages