On the Rate of Convergence of Weak Euler Approximation for Nondegenerate It\^{o} Diffusion and Jump Processes

TL;DR

This paper investigates how quickly the weak Euler approximation converges for nondegenerate Itô diffusion and jump processes with Hölder-continuous generators, including stable-driven SDEs, by analyzing solutions to the backward Kolmogorov equation.

Contribution

It establishes the convergence rate of the weak Euler scheme for a broad class of stochastic processes with Hölder-continuous generators, including nondegenerate diffusions and stable-driven SDEs.

Findings

Proves existence of unique solutions to the backward Kolmogorov equation in Hölder space.

Shows the Euler scheme has a positive weak order of convergence.

Applies results to processes driven by stable processes.

Abstract

The paper studies the rate of convergence of the weak Euler approximation for It\^{o} diffusion and jump processes with H\"{o}lder-continuous generators. It covers a number of stochastic processes including the nondegenerate diffusion processes and a class of stochastic differential equations driven by stable processes. To estimate the rate of convergence, the existence of a unique solution to the corresponding backward Kolmogorov equation in H\"{o}lder space is first proved. It then shows that the Euler scheme yields positive weak order of convergence.