Explicit approximations for nonlinear switching diffusion systems in finite and infinite horizons

TL;DR

This paper develops explicit numerical schemes to accurately approximate nonlinear switching diffusion systems driven by Markov chains, ensuring strong convergence and preservation of long-term dynamical properties in both finite and infinite horizons.

Contribution

It introduces easily implementable explicit schemes for nonlinear switching diffusions, guaranteeing convergence and long-term property preservation without restrictive conditions.

Findings

Numerical solutions converge strongly to exact solutions.

The schemes preserve stability, boundedness, and ergodicity.

Simulations validate theoretical results.

Abstract

Focusing on hybrid diffusion dynamics involving continuous dynamics as well as discrete events, this article investigates the explicit approximations for nonlinear switching diffusion systems modulated by a Markov chain. Different kinds of easily implementable explicit schemes have been proposed to approximate the dynamical behaviors of switching diffusion systems with local Lipschitz continuous drift and diffusion coefficients in both finite and infinite intervals. Without additional restriction conditions except those which guarantee the exact solutions posses their dynamical properties, the numerical solutions converge strongly to the exact solutions in finite horizon, moreover, realize the approximation of long-time dynamical properties including the moment boundedness, stability and ergodicity. Some simulations and examples are provided to support the theoretical results and demonstrate the validity of the approach.