About some skewed Brownian diffusions: explicit representation of their transition densities and exact simulation

In this talk we first discuss an explicit representation of the transition density of Brownian dynamics undergoing their motion through semipermeable and semireflecting barriers, called skewed Brownian motions.
We use this result to present an exact simulation of these diffusions, and comment some (still) open problems. Eventually we consider the exact simulation of Brownian diffusions whose drift admits finitely many jumps.

Clemens Printz (Universität Duisburg-Essen)
Stochastic averaging for multiscale Markov processes applied to a Wright-Fisher model with fluctuating

We present a new result on stochastic averaging for sequences of bivariate Markov processes $((X_t^n,Z_t^n)_{t\in[0,\infty)})_{n\in\mathbb{N}}$ whose components evolve on different time scales. Under suitable conditions, convergence of certain functionals of the fast variables $Z^n$ guarantees convergence of the (not necessarily Markovian on its own) slow variables $X^n$ to a limiting Markov process $(X_t)_{t\in[0,\infty)}$.
With this tool we can generalize the well-known diffusion limit of a Wright-Fisher model with randomly fluctuating selection. Whereas the classical result assumes the selection coefficients to be independent for different generations, we allow the environment to persist with a positive probability. The diffusion limit turns out to depend on this probability.
This talk is based on joint work with Martin Hutzenthaler and Peter Pfaffelhuber.