On multitype Branching Processes with Interaction

TL;DR

This paper introduces a new class of multitype branching processes with interactions, inspired by the stochastic Lotka-Volterra model, and establishes their connection to continuous-state processes via scaling limits and transformations.

Contribution

It develops a discrete and continuous framework for interacting multitype branching processes, including their construction, scaling limits, and relation to Le9vy processes, extending existing models.

Findings

Scaling limits of discrete processes match the continuous model

Continuous model constructed via generalized Lamperti transformation

Process can be represented as sum of random walk and Poisson process

Abstract

Motivated by the stochastic Lotka-Volterra model, we introduce discrete-state interacting multitype branching processes. We show that they can be obtained as the sum of a multidimensional random walk with a Lamperti-type change proportional to the population size; and a multidimensional Poisson process with a time-change proportional to the pairwise interactions. We define the analogous continuous-state process as the unique strong solution of a multidimensional SDE. We prove that the scaling limits of the discrete-state process correspond to its continuous counterpart. In addition, we show that the continuous-state model can be constructed as a generalized Lamperti-type transformation of multidimensional L\'evy processes.