Stochastic differential games with reflection and related obstacle problems for Isaacs equations |
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Authors: | Rainer Buckdahn Juan Li |
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Institution: | Rainer BUCKDAHN 1,3,Juan LI 2,1 Département de Mathématiques,Université de Bretagne Occidentale,6,avenue Victor-le-Gorgeu,B.P.809,29285 Brest cedex,France 2 School of Mathematics and Statistics,Shandong University at Weihai,Weihai 264209,China 3 Institute for Advanced Study,Shangdong University,Jinan 250100,China |
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Abstract: | In this paper we first investigate zero-sum two-player stochastic differential games with reflection, with the help of theory
of Reflected Backward Stochastic Differential Equations (RBSDEs). We will establish the dynamic programming principle for
the upper and the lower value functions of this kind of stochastic differential games with reflection in a straightforward
way. Then the upper and the lower value functions are proved to be the unique viscosity solutions to the associated upper
and the lower Hamilton-Jacobi-Bellman-Isaacs equations with obstacles, respectively. The method differs significantly from
those used for control problems with reflection, with new techniques developed of interest on its own. Further, we also prove
a new estimate for RBSDEs being sharper than that in the paper of El Karoui, Kapoudjian, Pardoux, Peng and Quenez (1997),
which turns out to be very useful because it allows us to estimate the L
p
-distance of the solutions of two different RBSDEs by the p-th power of the distance of the initial values of the driving forward equations. We also show that the unique viscosity solution
to the approximating Isaacs equation constructed by the penalization method converges to the viscosity solution of the Isaacs
equation with obstacle. |
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Keywords: | stochastic differential games value function reflected backward stochastic differential equations dynamic programming principle Isaacs equations with obstacles viscosity solution |
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