Pathwise convergence of a modular simulation scheme for hybrid SDEs with memory

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. The dependence of transitions on the discrete component’s own history, not just the Euclidean process, is the central departure from existing hybrid SDE models. For mH-SDEs driven by Lévy 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.