Jump Type Stochastic Differential Equations with Non-Lipschitz Coefficients: Non Confluence, Feller and Strong Feller Properties, and Exponential Ergodicity

TL;DR

This paper studies multidimensional jump stochastic differential equations with non-Lipschitz coefficients, establishing conditions for nonexplosion, pathwise uniqueness, non confluence, Feller properties, irreducibility, and exponential ergodicity, with applications to Levy processes.

Contribution

It provides new non-Lipschitz conditions ensuring well-posedness and ergodic behavior of jump SDEs, extending existing theory to more general coefficients and Levy-driven systems.

Findings

Established nonexplosion conditions for jump SDEs.

Proved pathwise uniqueness under non-Lipschitz conditions.

Derived conditions for exponential ergodicity.

Abstract

This paper considers multidimensional jump type stochastic differential equations with super linear growth and non-Lipschitz coefficients. After establishing a sufficient condition for nonexplosion, this paper presents sufficient non-Lipschitz conditions for pathwise uniqueness. The non confluence property for solutions is investigated. Feller and strong Feller properties under non-Lipschitz conditions are investigated via the coupling method. Sufficient conditions for irreducibility and exponential ergodicity are derived. As applications, this paper also studies multidimensional stochastic differential equations driven by L\'evy processes and presents a Feynman-Kac formula for L\'evy type operators.