A pathwise construction of Birth-Death-Swap systems leading to an averaging result in the presence of two timescales

This paper deals with the stochastic modeling of a general class of heterogeneous population dynamics structured by discrete subgroups. Such processes generalize classical multitype Birth- Death processes by allowing swap events, i.e. transfers from one subgroup to another. The variability of the environment is also included and the population evolution is not Markovian. We propose a new representation of the population based on its jump measure, characterized as a multivariate counting process with specific support conditions, and which together with the population defines a Birth-Death-Swap (BDS) system. We first prove a general result, on the construction by strong domination of multivariate counting processes solutions of stochastic differential equations driven by extended Poisson measures. Under weaker assumptions than usual, the existence of BDS systems strongly dominated by a Cox-Birth process is obtained. This pathwise comparison is the main tool to obtain tightness results in the second part of the paper. The BDS system in the presence of two timescales is then studied, when swap events occur at a faster timescale than demographic events. A general averaging result for the demographic counting process is proven. Classical averaging results obtained in the Markov case cannot be applied here, and in order to overcome this difficulty, we rely on the stable convergence of processes involved. This mode of convergence is particularly well-suited to our general framework. At the limit, the aggregated population become a Birth-Death process with averaged intensities, resulting from a non-trivial aggregation of the subgroups birth and death intensities