Stochastic equations of non-negative processes with jumps

TL;DR

This paper investigates stochastic equations governing non-negative processes with jumps, establishing existence, uniqueness, and comparison properties, and applies these results to Levy-driven equations and branching processes with immigration.

Contribution

It provides new existence and uniqueness results for stochastic equations with jumps, including non-Lipschitz cases, and extends to applications in Levy processes and branching models.

Findings

Proved existence and uniqueness of strong solutions.

Established comparison properties of solutions.

Applied results to Levy processes and branching processes.

Abstract

We study stochastic equations of non-negative processes with jumps. The existence and uniqueness of strong solutions are established under Lipschitz and non-Lipschitz conditions. The comparison property of two solutions are proved under suitable conditions. The results are applied to stochastic equations driven by one-sided Levy processes and those of continuous state branching processes with immigration.