15:30

Han Gan (Waikato University, NZ)
Stationary distribution approximations for two-island and seed bank models
Abstract: In this talk we will discuss two-island Wright-Fisher models which are used to model genetic frequencies and variation for subdivided populations. One of the key components of the model is the level of migration between the two islands. We show that as the population size increases, the appropriate approximation and limit for the stationary distribution of a two-island Wright-Fisher Markov chain depends on the level of migration. In a related seed bank model, individuals in one of the islands stay dormant rather than reproduce. We give analogous results for the seed bank model, compare and contrast the differences and examine the effect the seed bank has on genetic variability. Our results are derived from a new development of Stein's method for the two-island diffusion model and existing results for Stein's method for the Dirichlet distribution.

16:15

Paul Hager (HU Berlin)
Unified Signature Cumulants and Generalized Magnus Expansions
Abstract: Signature cumulants, defined as the tensor-logarithm of expected signatures of semimartingales, are seen to satisfy a fundamental functional relation. This equation, in a deterministic setting, contains Hausdorff’s differential equation, which itself underlies Magnus’ expansion. The (commutative) case of multivariate cumulants arises as another special case and yields a new Riccati-type relation valid for general semimartingales. Here, the accompanying expansion provides a new view on recent “diamond" and "martingale cumulants" (Alos et al ’17, Lacoin et al ’19., Friz et al. ’20) expansions. We will further discuss possible applications in finance.

17:00

Joaquin Fontbona (University of Chile)
Quantitative mean-field limit for interacting branching diffusions
Abstract: We establish an explicit rate of convergence for some systems of mean-field interacting diffusions with logistic binary branching, towards solutions of nonlinear evolution equations with non-local self-diffusion and logistic mass growth describing their large population limit. The proof relies on a novel coupling argument for binary branching diffusions based on optimal transport, allowing us to sharply mimic the trajectory of the interacting branching population by means of a system of independent particles with suitably distributed random space-time births. We are thus able to derive an optimal convergence rate, in the dual bounded-Lipschitz distance on finite measures, for the empirical measure of the population, from the known convergence rate in Wasserstein distance of empirical distributions of i.i.d. samples. Our approach and results extend propagation of chaos techniques and ideas from kinetic models, to stochastic systems of interacting branching populations, and appear to be new in this setting, even in the simple case of pure binary branching diffusions. Joint work with Felipe Muñoz-Hernandez.