Actuarial Seminar: Prof. Georgiy Shevchenko (Kyiv School of Economics)

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 $X^\theta = \bigl((1-\alpha)B^\theta+(X_0)^{1-\alpha} \bigr)^{1/(1-\alpha)}$, where for $\theta\in(-1,1)$, $B^\theta$ is the $\theta$-skew Brownian motion, and $(x)^{\gamma} = |x|^\gamma \operator name{sign} x$. 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.

Seminar organized by Prof. E. Hashorva