Existence and Stability of Solutions to Non-Lipschitz Stochastic Differential Equations Driven by L\'evy Noise

TL;DR

This paper establishes the existence, uniqueness, and mean-square stability of solutions to non-Lipschitz stochastic differential equations driven by Le9vy noise using a successive approximation method.

Contribution

It extends the analysis of SDEs driven by Le9vy noise to non-Lipschitz conditions, which are weaker than traditional Lipschitz assumptions.

Findings

Proves existence and uniqueness of solutions under non-Lipschitz conditions

Demonstrates mean-square stability of solutions

Uses successive approximation method for analysis

Abstract

In this paper, the successive approximation method is applied to investigate the existence and uniqueness of solutions to the stochastic differential equations (SDEs) driven by L\'evy noise under non-Lipschitz condition which is a much weaker condition than Lipschiz one. The stability of the solutions to non-Lipschitz SDEs driven by L\'evy noise is also considered, and the stochastic stability is obtained in the sense of mean square.