Stratonovich SDE with irregular coefficients: Girsanov's example revisited

Abstract

In this paper we study the Stratonovich stochastic differential equation $\mathrm{d} X=|X|^{\alpha}\circ\mathrm{d} B$, $\alpha\in(-1,1)$, which has been introduced by Cherstvy et al. [New Journal of Physics 15:083039 (2013)] in the context of analysis of anomalous diffusions in heterogeneous media. We determine its weak and strong solutions, which are homogeneous strong Markov processes \chng{spending zero time at $0$: for $\alpha\in (0,1)$, these solutions have the form $$ X_t^\theta=\bigl((1-\alpha)B_t^\theta\bigr)^{1/(1-\alpha)}, $$ where $B^\theta$ is the $\theta$-skew Brownian motion driven by $B$ and starting at $\frac{1}{1-\alpha}(X_0)^{1-\alpha}$, $\theta\in [-1,1]$,} and $(x)^{\gamma}=|x|^\gamma\operatorname{sign} x$; for $\alpha\in(-1,0]$, only the case $\theta=0$ is possible. The central part of the paper consists in the proof of the existence of a quadratic covariation $[f(B^\theta),B]$ for a locally square integrable function $f$ and is based on the time-reversion technique for Markovian diffusions.