Some Properties of Numerical Solutions for Semilinear Stochastic Delay Differential Equations Driven by G-Brownian Motion

Haiyan Yuan,Yi Qi

doi:10.1155/2021/1835490

Abstract

This paper is concerned with the numerical solutions of semilinear stochastic delay differential equations driven by G-Brownian motion (G-SLSDDEs). The existence and uniqueness of exact solutions of G-SLSDDEs are studied by using some inequalities and the Picard iteration scheme first. Then the numerical approximation of exponential Euler method for G-SLSDDEs is constructed, and the convergence and the stability of the numerical method are studied. It is proved that the exponential Euler method is convergent, and it can reproduce the stability of the analytical solution under some restrictions. Numerical experiments are presented to confirm the theoretical results.

Highlights

Many models in many branches of science and industry, such as economics, finance, biology, and medicine, reveal stochastic effects and are introduced as stochastic differential equations (SDEs)
Since most SDEs cannot be solved explicitly, numerical approximations which are on the basis of incorporating the stochastic factor in the classical numerical approximations for DDEs have become an important tool in the study of SDEs
E phenomenon of stiffness appears in the process of applying a certain numerical method to ODEs and SDEs

Summary

Introduction

Many models in many branches of science and industry, such as economics, finance, biology, and medicine, reveal stochastic effects and are introduced as stochastic differential equations (SDEs). Peng [20] gave the notions of G-expectation and G-Brownian motion on sublinear expectation space which provide the new perspective for the stochastic calculus under model uncertainty, which has aroused great interest. It can be found that most researches focus on linear and nonlinear SDEs, SDDEs, and NSDDEs with G-Brownian motion; there are a few numerical analysis results for semilinear SDDEs with G-Brownian motion (G-SLSDDEs). To fill this gap, we investigate the numerical solutions of G-SLSDDEs and give some results in the present paper.

Preliminaries

Exponential Euler Method and the Numerical Analysis

Numerical Examples