Hosts: prof. dr hab. Ryszard Szekli, prof. dr hab. Krzysztof Dębicki

Contact: dr Michał Krawiec (secretary), michal.krawiec at math.uni.wroc.pl

Current activity:

The seminar takes place once a week, on Thursdays from 12:15 - 14:00 in the form of a stationary meeting in room 603 (Mathematical Institute, UWr) or an online meeting on the Zoom platform. All current and more detailed information can be obtained from the seminar secretary.

Next seminar: 2025-01-30, 12:15 - 14:00

prof. Georgiy Shevchenko (Kyiv School of Economics)

Stratonovich stochastic differential equation with power non-linearity: (non)-uniqueness and selection problem

Abstract:
I will review results regarding a Stratonovich stochastic differential equation $$X_t=X_0+{\int }_0^t|X_s{|}^{\alpha }\circ d{B}_s,$$ which was introduced in the physical literature under the name ``heterogeneous diffusion process''. It turns out that equation has properties quite different from its Ito counterpart. Namely, we show that for $\alpha \in (0,1)$ it has infinitely many strong solutions spending zero time at zero. They are given by $X^{\theta }=((1-\alpha )B^{\theta }+(X_0{)}^{1-\alpha }{)}^{1/(1-\alpha )}$, where for $\theta \in (-1,1)$, $B^{\theta }$ is the $\theta$-skew Brownian motion, and $(x{)}^{\gamma }=|x{|}^{\gamma }\mathrm{sign}x$. It appears that there are no other homogeneous strong Markov solutions to the equation. To address the non-uniqueness, we consider a perturbation of the equation by a small independent noise. It appears that the solution to such equations converge to the solution of initial equation corresponding to $\theta =0$, i.e. the physically symmetric case.