Pathwise Convergence of a Modular Simulation Scheme for Hybrid SDEs with Memory

TL;DR

This paper introduces a new class of hybrid SDEs with memory, develops a modular simulation algorithm for them, and proves its pathwise convergence with explicit rates, enabling richer stochastic modeling.

Contribution

It extends hybrid SDE models to include joint history dependence and provides the first rigorous convergence analysis for such systems with jump-diffusion dynamics.

Findings

Developed the Modular-Poisson algorithm for mH-SDEs driven by Lévy processes.

Proved pathwise convergence with explicit error rates.

Established techniques to control error accumulation across regime changes.

Abstract

This work introduces hybrid stochastic differential equations with memory (mH-SDEs), a new class of stochastic systems where transition rates depend on the joint history of both Euclidean and discrete components. This extends existing hybrid stochastic differential equation models that condition transitions only on the Euclidean process history, enabling richer dependencies such as age-based transitions and self-reinforcing dynamics. For mH-SDEs driven by L\'evy processes, we develop the Modular-Poisson algorithm, which employs path-dependent uniformization to generate discrete jumps while advancing the Euclidean evolution between jumps using any established SDE solver as a micro-algorithm. The main theoretical contribution establishes pathwise convergence with explicit rates, developing new techniques to control error accumulation across regime changes and bound the probability of process decoupling. The modular design allows practitioners to employ existing, well-studied SDE simulation methods while preserving theoretical convergence properties. This work provides the first rigorous convergence analysis for hybrid systems with joint history dependence under jump-diffusion dynamics.