Ergodicity of stochastic functional differential equation with jumps and finite delay

TL;DR

This paper proves the ergodicity of certain stochastic differential equations with jumps and delays using coupling and support theorems, advancing understanding of their long-term behavior.

Contribution

It introduces a novel approach combining exponential decay bounds, Girsanov theorem, and support theorems to establish ergodicity for these complex equations.

Findings

Established ergodicity under specified conditions

Developed exponential decay bounds for coupled processes

Extended support theorems to the main process

Abstract

This paper investigates the ergodicity of stochastic functional differential equations with jumps under the Wasserstein distance by the generalized coupling method. Two key conditions are verified. The first is verified by establishing an exponential decay bound for the coupled segment processes and applying the Girsanov theorem for It\^o-L\'evy processes. The second is verified through a support theorem developed for an auxiliary process and then extended to the underlying process. Combining these results yields the desired ergodicity.