## Bernoulli

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

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

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

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

Digital Object Identifier
doi:10.3150/19-BEJ1161

Mathematical Reviews number (MathSciNet)
MR4058371

Zentralblatt MATH identifier
07166567

#### 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

#### References

• [1] Appuhamillage, T., Bokil, V., Thomann, E., Waymire, E. and Wood, B. (2011). Occupation and local times for skew Brownian motion with applications to dispersion across an interface. Ann. Appl. Probab. 21 183–214.
• [2] Aryasova, O.V. and Pilipenko, A.Y. (2011). On the strong uniqueness of a solution to singular stochastic differential equations. Theory Stoch. Process. 17 1–15.
• [3] Barlow, M., Burdzy, K., Kaspi, H. and Mandelbaum, A. (2000). Variably skewed Brownian motion. Electron. Commun. Probab. 5 57–66.
• [4] Bass, R.F., Burdzy, K. and Chen, Z.-Q. (2007). Pathwise uniqueness for a degenerate stochastic differential equation. Ann. Probab. 35 2385–2418.
• [5] Beck, A. (1973). Uniqueness of flow solutions of differential equations. In Recent Advances in Topological Dynamics (Proc. Conf. Topological Dynamics, Yale Univ., New Haven, Conn., 1972; in Honor of Gustav Arnold Hedlund). Lecture Notes in Math. 318 30–50. Berlin: Springer.
• [6] Cherny, A.S. (2001). Principal values of the integral functionals of Brownian motion: Existence, continuity and an extension of Itô’s formula. In Séminaire de Probabilités, XXXV. Lecture Notes in Math. 1755 348–370. Berlin: Springer.
• [7] Cherny, A.S. and Engelbert, H.-J. (2005). Singular Stochastic Differential Equations. Lecture Notes in Math. 1858. Berlin: Springer.
• [8] Cherstvy, A.G., Chechkin, A.V. and Metzler, R. (2013). Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes. New J. Phys. 15 083039, 13.
• [9] Denisov, S.I. and Horsthemke, W. (2002). Exactly solvable model with an absorbing state and multiplicative colored Gaussian noise. Phys. Rev. E (3) 65 061109, 10.
• [10] Dynkin, E.B. (1965). Markov Processes. Berlin: Springer.
• [11] Engelbert, H.-J. and Schmidt, W. (1985). On solutions of one-dimensional stochastic differential equations without drift. Z. Wahrsch. Verw. Gebiete 68 287–314.
• [12] Engelbert, H.J. and Schmidt, W. (1981). On the behaviour of certain functionals of the Wiener process and applications to stochastic differential equations. In Stochastic Differential Systems (Visegrád, 1980). Lecture Notes in Control and Information Sciences 36 47–55. Berlin: Springer.
• [13] Étoré, P. and Martinez, M. (2012). On the existence of a time inhomogeneous skew Brownian motion and some related laws. Electron. J. Probab. 17 no. 19, 27.
• [14] Föllmer, H., Protter, P. and Shiryayev, A.N. (1995). Quadratic covariation and an extension of Itô’s formula. Bernoulli 1 149–169.
• [15] Gairat, A. and Shcherbakov, V. (2017). Density of skew Brownian motion and its functionals with application in finance. Math. Finance 27 1069–1088.
• [16] Girsanov, I.V. (1962). An example of non-uniqueness of the solution of the stochastic equation of K. Ito. Theory Probab. Appl. 7 325–331.
• [17] Harrison, J.M. and Shepp, L.A. (1981). On skew Brownian motion. Ann. Probab. 9 309–313.
• [18] Haussmann, U.G. and Pardoux, É. (1985). Time reversal of diffusion processes. In Stochastic Differential Systems (Marseille–Luminy, 1984). Lect. Notes Control Inf. Sci. 69 176–182. Berlin: Springer.
• [19] Haussmann, U.G. and Pardoux, É. (1986). Time reversal of diffusions. Ann. Probab. 14 1188–1205.
• [20] Keilson, J. and Wellner, J.A. (1978). Oscillating Brownian motion. J. Appl. Probab. 15 300–310.
• [21] Lejay, A. (2006). On the constructions of the skew Brownian motion. Probab. Surv. 3 413–466.
• [22] Lejay, A. and Pigato, P. (2018). Statistical estimation of the oscillating Brownian motion. Bernoulli 24 3568–3602.
• [23] McKean, H.P. Jr. (1969). Stochastic Integrals. Probability and Mathematical Statistics 5. New York: Academic Press.
• [24] Pavlyukevich, I. and Shevchenko, G. (2020). Supplement to “Stratonovich stochastic differential equation with irregular coefficients: Girsanov’s example revisited.” https://doi.org/10.3150/19-BEJ1161SUPP.
• [25] Petit, F. (1997). Time reversal and reflected diffusions. Stochastic Process. Appl. 69 25–53.
• [26] Protter, P.E. (2004). Stochastic Integration and Differential Equations, 2nd ed. Applications of Mathematics (New York) 21. Berlin: Springer.
• [27] Russo, F. and Vallois, P. (1993). Forward, backward and symmetric stochastic integration. Probab. Theory Related Fields 97 403–421.
• [28] Russo, F. and Vallois, P. (1995). The generalized covariation process and Itô formula. Stochastic Process. Appl. 59 81–104.
• [29] Russo, F. and Vallois, P. (2000). Stochastic calculus with respect to continuous finite quadratic variation processes. Stoch. Stoch. Rep. 70 1–40.
• [30] Varadhan, S.R.S. (2011). Chapter 16. Reflected Brownian motion. Available at https://math.nyu.edu/~varadhan/Spring11/topics16.pdf.
• [31] Weinryb, S. (1983). Étude d’une équation différentielle stochastique avec temps local. In Seminar on Probability, XVII. Lecture Notes in Math. 986 72–77. Berlin: Springer.
• [32] Zvonkin, A.K. (1974). A transformation of the phase space of a diffusion process that will remove the drift. Math. USSR, Sb. 22 129.

#### Supplemental materials

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