Stochastic Target Games and Dynamic Programming via Regularized Viscosity Solutions

Bruno Bouchard () and Marcel Nutz ()
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Bruno Bouchard: CREST - Centre de Recherche en Économie et Statistique - ENSAI - Ecole Nationale de la Statistique et de l'Analyse de l'Information [Bruz] - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - ENSAE Paris - École Nationale de la Statistique et de l'Administration Économique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique, CEREMADE - CEntre de REcherches en MAthématiques de la DEcision - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CNRS - Centre National de la Recherche Scientifique
Marcel Nutz: Dept. of Mathematics - Columbia University [New York]

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Abstract: 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 mea-surable 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 ver-ification arguments for classical solutions of Hamilton–Jacobi–Bellman equations. The smooth supersolutions are constructed by an exten-sion 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.

Keywords: Stochastic differential game; Knightian uncertainty; 91B28; Shaking of coefficients; Viscosity solution AMS 2000 Subject Classification 93E20; 49L20; Stochastic target (search for similar items in EconPapers)
Date: 2015-07-10
New Economics Papers: this item is included in nep-dge, nep-gth and nep-ore
Note: View the original document on HAL open archive server: https://hal.science/hal-00846830v2
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Published in Mathematics of Operations Research, 2015, ⟨10.1287/moor.2015.0718⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-00846830

DOI: 10.1287/moor.2015.0718

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