Strong Predictor-Corrector Euler-Maruyama Methods for Stochastic Differential Equations with Markovian Switching

TL;DR

This paper introduces strong predictor-corrector Euler-Maruyama methods for stochastic differential equations with Markovian switching, improving accuracy and stability in pathwise simulations.

Contribution

It develops a new family of numerical methods that enhance error control and convergence analysis for SDEs with Markovian switching.

Findings

Proves strong convergence of the methods.

Establishes the order of the error under certain conditions.

Demonstrates computational efficiency through numerical examples.

Abstract

In this paper numerical methods for solving stochastic differential equations with Markovian switching (SDEwMSs) are developed by pathwise approximation. The proposed family of strong predictor-corrector Euler-Maruyama methods is designed to overcome the propagation of errors during the simulation of an approximate path. This paper not only shows the strong convergence of the numerical solution to the exact solution but also reveals the order of the error under some conditions on the coefficient functions. A natural analogue of $p$-stability criterion is studied. Numerical examples are given to illustrate the computational efficiency of the new predictor-corrector Euler-Maruyama approximation.