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August 2017 A strong order $1/2$ method for multidimensional SDEs with discontinuous drift
Gunther Leobacher, Michaela Szölgyenyi
Ann. Appl. Probab. 27(4): 2383-2418 (August 2017). DOI: 10.1214/16-AAP1262

Abstract

In this paper, we consider multidimensional stochastic differential equations (SDEs) with discontinuous drift and possibly degenerate diffusion coefficient. We prove an existence and uniqueness result for this class of SDEs and we present a numerical method that converges with strong order $1/2$. Our result is the first one that shows existence and uniqueness as well as strong convergence for such a general class of SDEs.

The proof is based on a transformation technique that removes the discontinuity from the drift such that the coefficients of the transformed SDE are Lipschitz continuous. Thus the Euler–Maruyama method can be applied to this transformed SDE. The approximation can be transformed back, giving an approximation to the solution of the original SDE.

As an illustration, we apply our result to an SDE the drift of which has a discontinuity along the unit circle and we present an application from stochastic optimal control.

Citation

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Gunther Leobacher. Michaela Szölgyenyi. "A strong order $1/2$ method for multidimensional SDEs with discontinuous drift." Ann. Appl. Probab. 27 (4) 2383 - 2418, August 2017. https://doi.org/10.1214/16-AAP1262

Information

Received: 1 July 2016; Published: August 2017
First available in Project Euclid: 30 August 2017

zbMATH: 1373.60102
MathSciNet: MR3693529
Digital Object Identifier: 10.1214/16-AAP1262

Subjects:
Primary: 60H10 , 65C20 , 65C30
Secondary: 65L20

Keywords: degenerate diffusion , discontinuous drift , existence and uniqueness of solutions , numerical methods for stochastic differential equations , Stochastic differential equations , strong convergence rate

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.27 • No. 4 • August 2017
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