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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 SDE. The CPS is a seamless integration of the SDE and the numerical method, unlike in the literature where they are treated as separate systems linked by inequalities. We formulate a new and general class of stochastic impulsive differential equations (SiDEs) that can serve as a canonic form of the CPSs and establish a Lyapunov stability theory as a theoretic foundation for our class of SiDEs. By the CPS approach, we show the equivalence and intrinsic relationship between the stability of the SDE and the stability of the numerical method. As application of our proposed results, we develop the CPS theory for linear systems and present the CPS Lyapunov inequality that is the necessary and sufficient condition for mean-square stability of the CPS of the Euler-Maruyama method for linear SDEs. Our proposed CPS theory initiates the study of systems numerics and provokes many open and interesting problems for future work.

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Stability of Cyber-Physical Systems of Numerical Methods for Stochastic Differential Equations: Integrating the Cyber and the Physical of Stochastic Systems