Stochastic Target Games and Dynamic Programming via Regularized Viscosity Solutions
Bouchard, Bruno; Nutz, Marcel (2016), Stochastic Target Games and Dynamic Programming via Regularized Viscosity Solutions, Mathematics of Operations Research, 41, 1, p. 109-124. 10.1287/moor.2015.0718
TypeArticle accepté pour publication ou publié
External document linkhttps://hal.archives-ouvertes.fr/hal-00846830
Journal nameMathematics of Operations Research
Institute of Management Sciences
MetadataShow full item record
Centre de Recherche en Économie et Statistique [CREST]
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)We study a class of stochastic target games where one player tries to find a strategy such that the state process almost-surely reaches a given target, no matter which action is chosen by the opponent. Our main result is a geometric dynamic programming principle which allows us to characterize the value function as the viscosity solution of a non-linear partial differential equation. Because abstract measurable selection arguments cannot be used in this context, the main obstacle is the construction of measurable almost-optimal strategies. We propose a novel approach where smooth supersolutions are used to define almost-optimal strategies of Markovian type, similarly as in verification arguments for classical solutions of Hamilton-Jacobi-Bellman equations. The smooth supersolutions are constructed by an extension of Krylov's method of shaken coefficients. We apply our results to a problem of option pricing under model uncertainty with different interest rates for borrowing and lending.
Subjects / KeywordsStochastic differential game; Viscosity solution; Stochastic target; Knightian uncertainty; Shaking of coefficients
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