Hybrid Stochastic Functional Differential Equations with Infinite Delay: Approximations and Numerics

TL;DR

This paper develops a theoretical framework showing that solutions to hybrid stochastic functional differential equations with infinite delay can be approximated by solutions with finite delay, enabling practical numerical methods.

Contribution

It introduces a new approximation theory for hybrid SFDEs with infinite delay, facilitating numerical solutions for highly nonlinear cases.

Findings

Positive approximation results for a broad class of nonlinear hybrid SFDEs

Enables numerical simulation of infinite delay SFDEs via finite delay models

Provides theoretical foundation for practical computational methods

Abstract

This paper is to investigate if the solution of a hybrid stochastic functional differential equation (SFDE) with infinite delay can be approximated by the solution of the corresponding hybrid SFDE with finite delay. A positive result is established for a large class of highly nonlinear hybrid SFDEs with infinite delay. Our new theory makes it possible to numerically approximate the solution of the hybrid SFDE with infinite delay, via the numerical solution of the corresponding hybrid SFDE with finite delay.