Birth Death Swap population in random environment and aggregation with two timescales

Abstract

This paper deals with the stochastic modeling of a class of heterogeneous population in a random environment, called birth-death-swap. In addition to demographic events, swap events, i.e. moves between subgroups, occur in the population. Event intensities are random functionals of the multi-type population. In the first part, we show that the complexity of the problem is significantly reduced by modeling the jumps measure of the population, described by a multivariate counting process. This process is defined as a solution of a stochastic differential system with random coefficients, driven by a multivariate Poisson random measure. The solution is obtained under weak assumptions, by the thinning of a strongly dominating point process generated by the same Poisson measure. This key construction relies on a general strong comparison result, of independent interest. The second part is dedicated to averaging results when swap events are significantly more frequent than demographic events. An important ingredient is the stable convergence, which is well-adapted to the general random environment. The pathwise construction by domination yields tightness results straightforwardly. At the limit, the demographic intensity functionals are averaged against random kernels depending on swap events. Finally, under a natural assumption, we show the convergence of the aggregated population to a "true" birth-death process in random environment, with non-linear intensity functionals.