Bernoulli

  • 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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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 \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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.bj/1580461583

Digital Object Identifier
doi:10.3150/19-BEJ1161

Mathematical Reviews number (MathSciNet)
MR4058371

Zentralblatt MATH identifier
07166567

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

Citation

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. https://projecteuclid.org/euclid.bj/1580461583


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Supplemental materials

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