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2015 On the existence of solutions of a class of SDEs with discontinuous drift and singular diffusion
Gunther Leobacher, Michaela Szölgyenyi, Stefan Thonhauser
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Electron. Commun. Probab. 20: 1-14 (2015). DOI: 10.1214/ECP.v20-3149


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.


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Gunther Leobacher. Michaela Szölgyenyi. Stefan Thonhauser. "On the existence of solutions of a class of SDEs with discontinuous drift and singular diffusion." Electron. Commun. Probab. 20 1 - 14, 2015.


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

zbMATH: 1308.65013
MathSciNet: MR3304412
Digital Object Identifier: 10.1214/ECP.v20-3149

Primary: 60H10

Keywords: degenerate diffusion , discontinuous drift , Stochastic differential equations

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