Actuarial Science Seminar: Stratonovich stochastic differential equation with power nonlinearity: Girsanov's example revisited

The talk is devoted to Stratonovich stochastic differential equation wherein the diffusion coefficient is a power function. It turns out that this equation, which was recently introduced in the physical literature by Cherstvy, Chechkin, and Metzler (2013) as heterogeneous diffusion process, has properties quite different from its Ito counterpart. Namely, we show that it has infinitely many strong solutions spending zero time at zero, they are given by transformations of skew Brownian motion. We moreover prove that there are no other homogeneous strong Markov solutions to the equation. The proofs use the time reversion technique for diffusions, which in the case of singular drifts leads to some peculiar effects. The presentation is based on joint research with Ilya Pavlyukevich (Friedrich Schiller University of Jena).