# Sample paths of continuous-state branching processes with dependent immigration

## TL;DR

This paper establishes the existence and uniqueness of sample paths for a class of continuous-state branching processes with dependent immigration, using stochastic integral equations driven by Poisson measures, under minimal moment assumptions.

## Contribution

It provides a direct construction of sample paths for these processes with dependent immigration, relaxing previous assumptions and allowing for model variations like competition.

## Key findings

- Proves existence and uniqueness of solutions to the stochastic integral equation.
- Constructs sample paths for processes with dependent immigration.
- Allows modifications in branching mechanisms and models with competition.

## Abstract

We prove the existence and pathwise uniqueness of the solution to a stochastic integral equation driven by Poisson random measures based on Kuznetsov measures for a continuous-state branching process. That gives a direct construction of the sample path of a continuous-state branching process with dependent immigration. The immigration rates depend on the population size via some functions satisfying a Yamada--Watanabe type condition. We only assume the existence of the first moment of the process. The existence of excursion law for the continuous-state branching process is not required. By special choices of the ingredients, we can make changes in the branching mechanism or construct models with competition.