Stochastic differential equation with jumps for multi-type continuous state and continuous time branching processes with immigration

TL;DR

This paper models multi-type continuous state and time branching processes with immigration using stochastic differential equations with jumps, providing a rigorous mathematical framework for such complex stochastic systems.

Contribution

It establishes the existence and uniqueness of solutions to the SDEs representing these processes, advancing the theoretical understanding of multi-type branching processes.

Findings

Proves pathwise uniqueness of solutions

Identifies the process as a strong solution of the SDEs

Provides conditions under which the model is well-defined

Abstract

A multi-type continuous state and continuous time branching process with immigration satisfying some moment conditions is identified as a pathwise unique strong solution of certain stochastic differential equation with jumps.