The weak rate of convergence for the Euler-Maruyama approximation of one-dimensional stochastic differential equations involving the local times of the unknown process

TL;DR

This paper investigates the weak convergence rate of the Euler-Maruyama method for one-dimensional stochastic differential equations that involve local times, using transformations to handle the local time component.

Contribution

It introduces a transformation technique to remove local times from SDEs, enabling analysis of Euler-Maruyama approximation for such equations and establishing convergence rates.

Findings

Established the weak convergence rate for Euler-Maruyama approximation.

Provided a transformation method to handle local times in SDEs.

Demonstrated convergence results for functions in a specific class.

Abstract

In this paper, we consider the weak convergence of the Euler-Maruyama approximation for one dimensional stochastic differential equations involving the local times of the unknown process. We use a transformation in order to remove the local time from the stochastic differential equations and we provide the approximation of Euler-maruyama for the stochastic differential equations without local time. After that, we conclude the approximation of Euler-maruyama for one dimensional stochastic differential equations involving the local times of the unknown process , and we provide the rate of weak convergence for any function G in a certain class.