# Development of numerical methods for nonlinear hybrid stochastic functional differential equations with infinite delay

## TL;DR

This paper develops and analyzes explicit numerical schemes for nonlinear hybrid stochastic functional differential equations with infinite delay, ensuring convergence, stability, and practical implementation with finite memory.

## Contribution

It introduces a finite-memory explicit scheme for complex stochastic equations with infinite delay and proves its boundedness, convergence rate, and stability properties.

## Key findings

- Proposed an explicit scheme requiring finite historical data.
- Proved the scheme's convergence with a rate of 1/2.
- Validated theoretical results through numerical experiments.

## Abstract

This paper addresses the challenging numerical simulation of nonlinear hybrid stochastic functional differential equations with infinite delays. We first propose an explicit scheme using space and time truncation, requiring only finite historical storage. Leveraging approximation theory, we prove the boundedness of the numerical solution's $p$th moment and establish its convergence, achieving a rate of $1/2$ order under polynomially growing coefficients. Furthermore, we refine the scheme to better capture the underlying exponential stability of the exact solution, in both moment and almost sure senses. Finally, numerical experiments are presented to validate our theoretical results.