Stratonovich SDE with irregular coefficients: Girsanov's example revisited

Abstract

In this paper we study the Stratonovich stochastic differential equation $\mathrm{d} X=|X|^α\circ\mathrm{d} B$, $α\in(-1,1)$, 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 $α\in (0,1)$, these solutions have the form $$ X_t^θ=\bigl((1-α)B_t^θ\bigr)^{1/(1-α)}, $$ where $B^θ$ is the $θ$-skew Brownian motion driven by $B$ and starting at $\frac{1}{1-α}(X_0)^{1-α}$, $θ\in [-1,1]$,} and $(x)^γ=|x|^γ\operatorname{sign} x$; for $α\in(-1,0]$, only the case $θ=0$ is possible. The central part of the paper consists in the proof of the existence of a quadratic covariation $[f(B^θ),B]$ for a locally square integrable function $f$ and is based on the time-reversion technique for Markovian diffusions.