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 $$X_t^{\theta }=((1-\alpha )B_t^{\theta }{)}^{1/(1-\alpha )},$$ 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 $(x{)}^{\gamma }=|x{|}^{\gamma }\mathrm{sign}x$; 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.