Open Access
May 2020 Stratonovich stochastic differential equation with irregular coefficients: Girsanov’s example revisited
Ilya Pavlyukevich, Georgiy Shevchenko
Bernoulli 26(2): 1381-1409 (May 2020). DOI: 10.3150/19-BEJ1161


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 \begin{equation*}X_{t}^{\theta }=((1-\alpha)B_{t}^{\theta })^{1/(1-\alpha )},\end{equation*} 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.


Download Citation

Ilya Pavlyukevich. Georgiy Shevchenko. "Stratonovich stochastic differential equation with irregular coefficients: Girsanov’s example revisited." Bernoulli 26 (2) 1381 - 1409, May 2020.


Received: 1 April 2019; Revised: 1 September 2019; Published: May 2020
First available in Project Euclid: 31 January 2020

zbMATH: 07166567
MathSciNet: MR4058371
Digital Object Identifier: 10.3150/19-BEJ1161

Keywords: generalized Itô’s formula , Girsanov’s example , heterogeneous diffusion process , Local time , non-uniqueness , singular stochastic differential equation , skew Brownian motion , Stratonovich integral , time reversion

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

Vol.26 • No. 2 • May 2020
Back to Top