# Stability of cyber-physical systems of numerical methods for stochastic differential equations: integrating the cyber and the physical of stochastic systems

## TL;DR

This paper introduces a unified cyber-physical system framework for stochastic differential equations and numerical methods, establishing a fundamental stability theory linking the stability of the SDE and its numerical approximation.

## Contribution

It develops a general class of stochastic impulsive differential equations to model CPSs of numerical methods, providing a new theoretical foundation for stability analysis.

## Key findings

- CPS framework unifies SDEs and numerical methods
- Lyapunov stability theory applied to CPSs
- Necessary and sufficient stability conditions for Euler-Maruyama

## Abstract

This paper presents the cyber-physical system (CPS) of a numerical method (the widely-used Euler- Maruyama method) and establishes a foundational theory of the CPSs of numerical methods for stochastic differential equations (SDEs), which transforms the way we understand the relationship between the numerical method and the underlying dynamical system. Unlike in the literature where they are treated as separate systems linked by inequalities, the CPS is a seamless integration of the SDE and the numerical method and we construct a new and general class of stochastic impulsive differential equations (SiDEs) that can serve as a canonic form of the CPSs of numerical methods. By the CPS approach, we show the equivalence and intrinsic relationship between the stability of the SDE and the stability of the numerical method using the Lyapunov stability theory we develop for our class of SiDEs. Applying our established theory, we present the CPS Lyapunov inequality that is the necessary and sufficient condition for meansquare stability of the CPS of the Euler-Maruyama method for linear SDEs. The proposed CPS and theory initiate the study of systems numerics and provoke many open and interesting problems for future work.