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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

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Type
Article accepté pour publication ou publié
External document link
https://hal.archives-ouvertes.fr/hal-00846830
Date
2016
Journal name
Mathematics of Operations Research
Volume
41
Number
1
Publisher
Institute of Management Sciences
Published in
Paris
Pages
109-124
Publication identifier
10.1287/moor.2015.0718
Metadata
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Author(s)
Bouchard, Bruno
Centre de Recherche en Économie et Statistique [CREST]
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Nutz, Marcel
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 / Keywords
Stochastic differential game; Viscosity solution; Stochastic target; Knightian uncertainty; Shaking of coefficients

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