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Stochastic Target Games and Dynamic Programming via Regularized Viscosity Solutions

Author

Listed:
  • 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 - 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])

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.

Suggested Citation

  • Bruno Bouchard & Marcel Nutz, 2015. "Stochastic Target Games and Dynamic Programming via Regularized Viscosity Solutions," Post-Print hal-00846830, HAL.
    • Handle: RePEc:hal:journl:hal-00846830
    DOI: 10.1287/moor.2015.0718
    Note: View the original document on HAL open archive server: https://hal.science/hal-00846830v2
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    References listed on IDEAS

    as
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    Cited by:

    1. Erhan Bayraktar & Jiaqi Li, 2016. "Stochastic Perron for Stochastic Target Problems," Journal of Optimization Theory and Applications, Springer, vol. 170(3), pages 1026-1054, September.
    2. Romuald Elie & Ludovic Moreau & Dylan Possamai, 2017. "On a class of path-dependent singular stochastic control problems," Papers 1701.08861, arXiv.org, revised Feb 2018.

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    More about this item

    Keywords

    Stochastic differential game; Knightian uncertainty; 91B28; Shaking of coefficients; Viscosity solution AMS 2000 Subject Classification 93E20; 49L20; Stochastic target;
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