A note on existence of global solutions and invariant measures for jump SDEs with locally one-sided Lipschitz drift

TL;DR

This paper extends existing methods to prove the existence of global strong solutions and invariant measures for jump stochastic differential equations with locally one-sided Lipschitz drift and local Lipschitz coefficients, under growth conditions.

Contribution

It introduces a novel application of methods to establish global solutions and invariant measures for jump SDEs with specific local Lipschitz and growth conditions.

Findings

Proved existence of global strong solutions under local one-sided Lipschitz conditions.

Established existence of invariant measures for a broad class of jump SDEs.

Extended methods from previous work to jump SDEs with monotonicity conditions.

Abstract

We extend some methods developed by Albeverio, Brze\'{z}niak and Wu and we show how to apply them in order to prove existence of global strong solutions of stochastic differential equations with jumps, under a local one-sided Lipschitz condition on the drift (also known as a monotonicity condition) and a local Lipschitz condition on the diffusion and jump coefficients, while an additional global one-sided linear growth assumption is satisfied. Then we use these methods to prove existence of invariant measures for a broad class of such equations.