Stability of stochastic dynamic systems of a random structure with Markov switching in the presence of concentration points

This article aims to investigate sufficient conditions for the stability of the trivial solution of stochastic differential equations with a random structure, particularly in contexts involving the presence of concentration points. The proof of asymptotic stability leverages the use of Lyapunov functions, supplemented by additional constraints on the magnitudes of jumps and jump times, as well as the Markov property of the system solutions. The findings are elucidated with an example, demonstrating both stable and unstable conditions of the system. The novelty of this work is in the consideration of jump concentration points, which are not considered in classical works. The assumption of the existence of concentration points leads to additional constraints on jumps, jump times and relations between them.

Automated summary

This article investigates the stability of trivial solutions of stochastic differential equations with a random structure, particularly when concentration points are present. The proof of asymptotic stability relies on Lyapunov functions, additional constraints on jumps and jump times, and the Markov property. An example is provided to illustrate both stable and unstable conditions of the system. The novelty of this work lies in considering jump concentration points, which is not typically considered in classical works.