An averaging result for heterogeneous Birth-Death-Swap population dynamics in random environment

In this talk, I will present a class of heterogeneous population models 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, I will 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.
I will then present an 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. At the limit, the demographic intensity functionals are averaged against random kernels depending on swap events, and the aggregated population can be approximated by a "true" birth-death process in random environment.

Joint work with N. El Karoui.