First order convergence of weak Wong–Zakai approximations of Lévy-driven Marcus SDEs

Tetyana Kosenkova^a, Alexei Kulik^b, Ilya Pavlyukevich<sup>c

a</sup> Institute of Mathematics, University of Potsdam, Campus Golm, Karl--Liebknecht--Strasse 24--25, 14476 Potsdam, Germany
^b Wroclaw University of Science and Technology Faculty of Pure and Applied Mathematics, Wybrzeże Wyspiańskiego Str. 27, 50-370 Wroclaw, Poland
^c Institute of Mathematics, Friedrich Schiller University Jena, Ernst–Abbe–Platz 2, 07743 Jena, Germany

Abstract: For solutions $X=(X_t)_{t\in[0,T]}$ of a Lévy-driven Marcus (canonical) stochastic differential equation we study the Wong–Zakai type time discrete approximations $\bar X=(\bar X_{kh})_{0\leq k\leq T/h}$, $h>0$, and establish the first order convergence $|\mathbf{E}_x f(X_T)-\mathbf{E}_x f(X^h_T)|\leq C h$ for $f\in C_b^4$.

Keywords: Lévy process, Marcus (canonical) stochastic differential equation, Wong–Zakai approximation, first order convergence, Euler scheme.

MSC: 65C30, 60H10, 60G51 , 60H35

Language: English