Jump stochastic differential equations with non-Lipschitz and superlinearly growing coefficients

TL;DR

This paper investigates the existence, uniqueness, and properties of solutions to jump stochastic differential equations with super-linear, non-Lipschitz coefficients, providing new comparison results and applications to equations with logarithmic growth.

Contribution

It establishes conditions for no-explosion, pathwise uniqueness, and continuity of solutions for jump SDEs with challenging coefficient behaviors, including a specific application to coefficients like x log|x|.

Findings

Proved no-explosion and pathwise uniqueness under super-linear growth conditions.

Established a comparison theorem for one-dimensional jump SDEs.

Showed existence of a unique strong solution for SDEs with coefficients like x log|x|.

Abstract

In the paper, we consider the no-explosion condition and pathwise uniqueness for SDEs driven by a Poisson random measure with coefficients that are super-linear and non-Lipschitz. We give a comparison theorem in the one-dimensional case under some additional condition. Moreover, we study the non-contact property and the continuity with respect to the space variable of the stochastic flow. As an application, we will show that there exists a unique strong solution for SDEs with coefficients like $x\log|x|$.