Uniqueness Problem for the Backward Differential Equation of a Continuous-State Branching Process

TL;DR

This paper proves the uniqueness of solutions to the backward differential equation governing the distributional properties of multi-dimensional continuous-state branching processes, clarifying foundational aspects of their mathematical characterization.

Contribution

It provides a rigorous proof of the uniqueness of solutions to the backward differential equation, confirming a longstanding assertion and characterizing the process via stochastic equations.

Findings

Confirmed the uniqueness of solutions to the backward differential equation.

Characterized the process as the pathwise unique solution to stochastic equations.

Validated the foundational properties of multi-dimensional continuous-state branching processes.

Abstract

The distributional properties of a multi-dimensional continuous-state branching process are determined by its cumulant semigroup, which is defined by the backward differential equation. We provide a proof of the assertion of Rhyzhov and Skorokhod (Theory Probab. Appl., 1970) on the uniqueness of the solutions to the equation, which is based on a characterization of the process as the pathwise unique solution to a system of stochastic equations.