Electronic Communications in Probability

On the existence of solutions of a class of SDEs with discontinuous drift and singular diffusion

Gunther Leobacher, Michaela Szölgyenyi, and Stefan Thonhauser

Full-text: Open access


The classical result by Itô on the existence of strong solutions of stochastic differential equations (SDEs) with Lipschitz coefficients can be extended to the case where the drift is only measurable and bounded. These generalizations are based on techniques presented by Zvonkin and Veretennikov, which rely on the uniform ellipticity of the diffusion coefficient.In this paper we study the case of degenerate ellipticity and give sufficient conditions for the existence of a solution. The conditions on the diffusion coefficient are more general than previous results and we gain fundamental insight into the geometric properties of the discontinuity of the drift on the one hand and the diffusion vector field on the other hand. Besides presenting existence results for the degenerate elliptic situation, we give an example illustrating the difficulties in obtaining more general results than those given.The particular types of SDEs considered arise naturally in the framework of combined optimal control and filtering problems.

Article information

Electron. Commun. Probab., Volume 20 (2015), paper no. 6, 14 pp.

Accepted: 15 January 2015
First available in Project Euclid: 7 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]

stochastic differential equations degenerate diffusion discontinuous drift

This work is licensed under a Creative Commons Attribution 3.0 License.


Leobacher, Gunther; Szölgyenyi, Michaela; Thonhauser, Stefan. On the existence of solutions of a class of SDEs with discontinuous drift and singular diffusion. Electron. Commun. Probab. 20 (2015), paper no. 6, 14 pp. doi:10.1214/ECP.v20-3149. https://projecteuclid.org/euclid.ecp/1465320933

Export citation


  • Eisenbaum, Nathalie. Integration with respect to local time. Potential Anal. 13 (2000), no. 4, 303–328.
  • Föllmer, Hans; Protter, Philip. On Itô's formula for multidimensional Brownian motion. Probab. Theory Related Fields 116 (2000), no. 1, 1–20.
  • Halidias, Nikolaos; Kloeden, P. E. A note on strong solutions of stochastic differential equations with a discontinuous drift coefficient. J. Appl. Math. Stoch. Anal. 2006, Art. ID 73257, 6 pp.
  • Karatzas, Ioannis; Shreve, Steven E. Brownian motion and stochastic calculus. Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. xxiv+470 pp. ISBN: 0-387-97655-8
  • Krylov, N. V.; Liptser, R. On diffusion approximation with discontinuous coefficients. Stochastic Process. Appl. 102 (2002), no. 2, 235–264.
  • Leobacher, Gunther; Szölgyenyi, Michaela; Thonhauser, Stefan. Bayesian dividend optimization and finite time ruin probabilities. Stoch. Models 30 (2014), no. 2, 216–249.
  • Mao, Xuerong. Stochastic differential equations and applications. Second edition. Horwood Publishing Limited, Chichester, 2008. xviii+422 pp. ISBN: 978-1-904275-34-3
  • Mao, Xuerong; Yuan, Chenggui. Stochastic differential equations with Markovian switching. Imperial College Press, London, 2006. xviii+409 pp. ISBN: 1-86094-701-8
  • Meyer-Brandis, Thilo; Proske, Frank. Construction of strong solutions of SDE's via Malliavin calculus. J. Funct. Anal. 258 (2010), no. 11, 3922–3953.
  • Protter, Philip E. Stochastic integration and differential equations. Second edition. Applications of Mathematics (New York), 21. Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2004. xiv+415 pp. ISBN: 3-540-00313-4
  • Rudin, Walter. Principles of mathematical analysis. Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976. x+342 pp.
  • Russo, F.; Vallois, P. Itô formula for $C^ 1$-functions of semimartingales. Probab. Theory Related Fields 104 (1996), no. 1, 27–41.
  • Veretennikov, A. Ju. Strong solutions and explicit formulas for solutions of stochastic integral equations. (Russian) Mat. Sb. (N.S.) 111(153) (1980), no. 3, 434–452, 480.
  • Veretennikov, A. Ju. On Stochastic Equations with Degenerate Diffusion with Respect to Some of the Variables, Mathematics of the USSR Izvestiya 22 (1984), no. 1, 173–180.
  • Zhang, Xicheng. Strong solutions of SDES with singular drift and Sobolev diffusion coefficients. Stochastic Process. Appl. 115 (2005), no. 11, 1805–1818.
  • Zvonkin, A. K. A transformation of the phase space of a diffusion process that will remove the drift. (Russian) Mat. Sb. (N.S.) 93(135) (1974), 129–149, 152.