Stratonovich stochastic differential equation with irregular coefficients: Girsanov’s example revisited
Ilya Pavlyukevich, Georgiy Shevchenko.
Source: Bernoulli, Volume 26, Number 2, 1381--1409.
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 J. Phys. 15 (2013) 083039) in the context of analysis of anomalous diffusions in heterogeneous media. We determine its weak and strong solutions, which are homogeneous strong Markov processes spending zero time at $0$: for $alpha in (0,1)$, these solutions have the form egin{equation*}X_{t}^{ heta }=((1-alpha)B_{t}^{ heta })^{1/(1-alpha )},end{equation*} where $B^{ heta }$ is the $ heta $-skew Brownian motion driven by $B$ and starting at $frac{1}{1-alpha }(X_{0})^{1-alpha }$, $ heta in [-1,1]$, and $(x)^{gamma }=|x|^{gamma }operatorname{sign}x$; for $alpha in (-1,0]$, only the case $ heta =0$ is possible. The central part of the paper consists in the proof of the existence of a quadratic covariation $[f(B^{ heta }),B]$ for a locally square integrable function $f$ and is based on the time-reversion technique for Markovian diffusions.