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

Kontakt: dr Michał Krawiec (sekretarz), michal.krawiec at math.uni.wroc.pl

For English version please see: English version.

Aktualna działalność:

Seminarium odbywa się raz w tygodniu, w czwartki w godz. 12:15 - 14:00 stacjonarnie w sali 603 lub w formie wideokonferencji na platformie Zoom. Wszystkie aktualne i bardziej szczegółowe informacje można uzyskać u sekretarza seminarium. Pełen wykaz zaplanowanych referatów znajduje się w zakładce Referaty.

Najbliższy referat: 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

Streszczenie:
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.