Stochastic selection problem for a Stratonovich SDE with power non-linearity

Abstract

In our paper [Bernoulli 26(2), 2020, 1381-1409], we found all strong Markov solutions that spend zero time at $0$ of the Stratonovich stochastic differential equation $dX=|X{|}^{\alpha }\circ dB$, $\alpha \in (0,1)$. These solutions have the form $X_t^{\theta }=F(B_t^{\theta })$, where $F(x)=\frac{1}{1-\alpha }|x{|}^{1/(1-\alpha )}\text{sign}x$ and $B^{\theta }$ is the skew Brownian motion with skewness parameter $\theta \in [-1,1]$ starting at $F^{-1}(X_0)$. In this paper we show how an addition of small external additive noise $\epsilon W$ restores uniqueness. In the limit as $\epsilon \to 0$, we recover heterogeneous diffusion corresponding to the physically symmetric case $\theta =0$.

18 pages, 1 figure