Phase Aggregation and Poisson Approximation in Multi-Scale Markov-Switching Stochastic Dynamical Systems

Abstract

We study nonlinear stochastic dynamical systems that evolve in a Markov environment with separated fast and slow transition scales. These systems are also subject to impulsive perturbations under a Poisson approximation scheme. The environment is represented on a product state space, and phase aggregation is used to average the rapidly switching component while retaining the slower Markov regime explicitly. Under the stated ergodicity, regularity, integrability, tightness, state-preservation, and uniqueness assumptions, we derive an effective reduced process and establish weak convergence of both the impulsive component and the coupled nonlinear system in the Skorokhod space. The limiting jump dynamics are characterized by averaged local characteristics associated with the retained slow Markov regime. To assess the reduced model numerically, we use a nonlinear competitive Lotka–Volterra-type system and compare the deterministic dynamics, the full Markov-switching stochastic model, and the reduced approximation under several fixed parameter configurations and jump-intensity settings. The results provide a mathematical basis for the reduced modeling of nonlinear stochastic dynamical systems with separated Markov time scales and impulsive perturbations.