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Probabilistic interpretation for systems of Isaacs equations with two reflecting barriers

Content Provider	Springer Nature Link
Author	Buckdahn, Rainer Li, Juan
Copyright Year	2009
Abstract	In this paper we investigate zero-sum two-player stochastic differential games whose cost functionals are given by doubly controlled reflected backward stochastic differential equations (RBSDEs) with two barriers. For admissible controls which can depend on the whole past and so include, in particular, information occurring before the beginning of the game, the games are interpreted as games of the type “admissible strategy” against “admissible control”, and the associated lower and upper value functions are studied. A priori random, they are shown to be deterministic, and it is proved that they are the unique viscosity solutions of the associated upper and the lower Bellman–Isaacs equations with two barriers, respectively. For the proofs we make full use of the penalization method for RBSDEs with one barrier and RBSDEs with two barriers. For this end we also prove new estimates for RBSDEs with two barriers, which are sharper than those in Hamadène, Hassani (Probab Theory Relat Fields 132:237–264, 2005). Furthermore, we show that the viscosity solution of the Isaacs equation with two reflecting barriers not only can be approximated by the viscosity solutions of penalized Isaacs equations with one barrier, but also directly by the viscosity solutions of penalized Isaacs equations without barrier.
Ending Page	420
Page Count	40
Starting Page	381
File Format	PDF
ISSN	10219722
e-ISSN	14209004
Journal	Nonlinear Differential Equations and Applications NoDEA
Issue Number	3
Volume Number	16
Language	English
Publisher	SP Birkhäuser Verlag Basel
Publisher Date	2009-05-30
Publisher Place	Basel
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Dynamic programming principle Value function Isaacs equations with obstacles Viscosity solution Analysis Stochastic ordinary differential equations Dynamic programming Zero-sum games Reflected backward stochastic differential equations Stochastic differential games
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis

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