Euler-Maruyama method for stochastic differential equations with discontinuous drift

Description

When solving certain stochastic optimization problems, e.g., in mathematical finance, the optimal control policy sometimes turns out to be of threshold type, meaning that the control depends on the state of the controlled process in a discontinuous fashion. The stochastic differential equations (SDEs) modeling the underlying process then typically have discontinuous drift and degenerate diffusion parameter. This motivates the study of a more general class of such SDEs. We prove an existence and uniqueness result and present a numerical algorithm, both based on certain transformations of the state space. The transform is different from an earlier one by Zvonkin and Verettennikov in that the drift is not removed entirely by the transform, but is merely ``made continuous''. As a consequence the transform becomes computable without the necessity of solving systems of partial differential equations numerically. The resulting numerical method is then feasible and proven to converge with strong order $1/2$. This is the first result of this kind for this class of SDEs. We will first present the one-dimensional case and subsequently show how the ideas can be generalized to higher dimensions. There we find a nice geometrical interpretation of our weakened non-degeneracy condition.

Period	24 Oct 2015
Event title	unbekannt/unknown
Event type	Conference
Location	AustriaShow on map