## Electronic Communications in Probability

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

#### Abstract

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

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

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

https://projecteuclid.org/euclid.ecp/1465320933

Digital Object Identifier
doi:10.1214/ECP.v20-3149

Mathematical Reviews number (MathSciNet)
MR3304412

Zentralblatt MATH identifier
1308.65013

Rights

#### Citation

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

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