The Truncated Theta-EM Method for Nonlinear and Nonautonomous Hybrid Stochastic Differential Delay Equations with Poisson Jumps

Weifeng Wang,Shuaibin Gao,Junhao Hu,Lei Yan,Shiping Wen

doi:10.1155/2021/2882076

Abstract

In this paper, we study a class of nonlinear and nonautonomous hybrid stochastic differential delay equations with Poisson jumps (HSDDEwPJs). The convergence rate of the truncated theta-EM numerical solutions to HSDDEwPJs is investigated under given conditions. An example is shown to support our theory.

Highlights

Stochastic differential equations have been widely used in many fields and have attracted many scholars [1,2,3]
The surprising outbreak of COVID-19 has a huge impact on the world economy, especially on the stock market. erefore, stochastic differential equations with jumps considering continuous and discontinuous random effects have been investigated to analyze these situations [4,5,6,7]
Discrete Dynamics in Nature and Society of EM approximation solution to the true solution in probability under some weaker conditions was proved in [33]. e EM approximate solutions converge to the true solutions for stochastic differential delay equations with Poisson jumps and Markovian switching under local Lipschitz condition [34]. e convergence of EM method for stochastic differential delay equations with Poisson jumps and Markovian switching in the sense of L1-norm under one non-Lipschitz condition was discussed in [35]. e strong convergence between the true solutions and the numerical solutions to stochastic differential delay equations with Poisson jumps and Markovian switching was studied when the drift and diffusion coefficients are Taylor approximations [36]

Summary

Introduction

Stochastic differential equations have been widely used in many fields and have attracted many scholars [1,2,3]. E EM approximate solutions converge to the true solutions for stochastic differential delay equations with Poisson jumps and Markovian switching under local Lipschitz condition [34]. E convergence of EM method for stochastic differential delay equations with Poisson jumps and Markovian switching in the sense of L1-norm under one non-Lipschitz condition was discussed in [35]. E strong convergence between the true solutions and the numerical solutions to stochastic differential delay equations with Poisson jumps and Markovian switching was studied when the drift and diffusion coefficients are Taylor approximations [36]. We consider the nonlinear and nonautonomous hybrid stochastic differential delay equations with Poisson jumps of the form dx(t) f(t, x(t), x(t − τ), r(t))dt + g(t, x(t), x(t − τ), r(t))dB(t) + h(t, x(t), x(t − τ))r(t))dN(t), t ∈ [0, T], (4).

Convergence Rate

Example