The rate of Lp-convergence for the Euler-Maruyama method of the stochastic differential equations with Markovian switching

TL;DR

This paper establishes that the Euler-Maruyama method for stochastic differential equations with Markovian switching achieves an Lp-convergence rate of 1/2, improving upon previous results that capped at 1/p, using a novel approach that directly incorporates the Markov chain.

Contribution

The paper demonstrates that the Lp-convergence rate of the EM method for SDEs with Markovian switching can reach 1/2, using a new technique involving direct use of the Markov chain.

Findings

Lp-convergence rate of EM method is 1/2

Direct use of Markov chain improves convergence analysis

Method applicable to other SDEs with Markovian switching

Abstract

This work deals with the Euler-Maruyama (EM) scheme for stochastic differential equations with Markovian switching (SDEwMSs). We focus on the Lp-convergence rate (p is greater than or equal to 2) of the EM method given in this paper. As far as we know, the skeleton process of the Markov chain is used in the continuous numerical methods in most papers. By contrast, the continuous EM method in this paper is to use the Markov chain directly. To the best of our knowledge, there are only two papers that consider the rate of Lp-convergence, which is no more than 1/p (p is greater than or equal to 2) in these papers. The contribution of this paper is that the rate of Lp-convergence of the EM method can reach 1/2. We believe that the technique used in this paper to construct the EM method can also be used to construct other methods for SDEwMSs.