On the Exponential Stability of Switching-Diffusion Processes with Jumps

TL;DR

This paper investigates the pathwise stability of stochastic PDEs driven by switching-diffusion processes with jumps, providing new criteria that do not depend on moment stability and highlighting the effects of Markovian switching.

Contribution

It introduces novel stability criteria for such systems that are independent of moment stability and accounts for the impact of switching on stability.

Findings

Sample Lyapunov exponent is generally smaller than in Wiener-driven systems.

System can be pathwise exponentially stable due to switching, even if some subsystems are unstable.

Stability criteria do not rely on moment stability of the system.

Abstract

In this paper we focus on the pathwise stability of mild solutions for a class of stochastic partial differential equations which are driven by switching-diffusion processes with jumps. In comparison to the existing literature, we show that: (i) the criterion to guarantee pathwise stability does not rely on the moment stability of the system; (ii) the sample Lyapunov exponent obtained is generally smaller than that of the counterpart driven by a Wiener process; (iii) due to the Markovian switching the overall system can become pathwise exponentially stable although some subsystems are not stable.