Stochastic Differential Games and Viscosity Solutions of Hamilton-Jacobi-Bellman-Isaacs Equations

Details Stochastic Differential Games and Viscosity Solutions of Hamilton-Jacobi-Bellman-Isaacs Equations Stochastic Differential Games and Viscosity Solutions of Hamilton-Jacobi-Bellman-Isaacs Equations

2007-02-06

arxiv.org/abs/math/0702131v1

SIAM J.Contr.Opt. 47(1), 444-475,2008.

Mathematics

Probability

The results were presented by Rainer Buckdahn at the "12th International Symposium on Dynamic Games and Applications" in Sop

Scientific paper

10.1137/060671954

In this paper we study zero-sum two-player stochastic differential games with the help of theory of Backward Stochastic Differential Equations (BSDEs). At the one hand we generalize the results of the pioneer work of Fleming and Souganidis by considering cost functionals defined by controlled BSDEs and by allowing the admissible control processes to depend on events occurring before the beginning of the game (which implies that the cost functionals become random variables), on the other hand the application of BSDE methods, in particular that of the notion of stochastic "backward semigroups" introduced by Peng allows to prove a dynamic programming principle for the upper and the lower value functions of the game in a straight-forward way, without passing by additional approximations. The upper and the lower value functions are proved to be the unique viscosity solutions of the upper and the lower Hamilton-Jacobi-Bellman-Isaacs equations, respectively. For this Peng's BSDE method is translated from the framework of stochastic control theory into that of stochastic differential games.

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