• Bernoulli
  • Volume 26, Number 2 (2020), 1381-1409.

Stratonovich stochastic differential equation with irregular coefficients: Girsanov’s example revisited

Ilya Pavlyukevich and Georgiy Shevchenko

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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.

Article information

Bernoulli, Volume 26, Number 2 (2020), 1381-1409.

Received: April 2019
Revised: September 2019
First available in Project Euclid: 31 January 2020

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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


Pavlyukevich, Ilya; Shevchenko, Georgiy. Stratonovich stochastic differential equation with irregular coefficients: Girsanov’s example revisited. Bernoulli 26 (2020), no. 2, 1381--1409. doi:10.3150/19-BEJ1161.

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Supplemental materials

  • Supplement: Additional derivations. We provide technical proofs omitted from the main body of the article.