Probabilistic interpretation for systems of Isaacs equations with two reflecting barriers |
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Authors: | Rainer Buckdahn Juan Li |
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Institution: | (1) Département de Mathématiques, Université de Bretagne Occidentale, 6, avenue Victor-le-Gorgeu, CS 93837, 29238 Brest Cedex 3, France;(2) School of Mathematics and Statistics, Shandong University at Weihai, 264209 Weihai, People’s Republic of China |
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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.
Partially supported by the NSF of P.R.China (No. 10701050; 10671112), Shandong Province (No. Q2007A04), and National Basic
Research Program of China (973 Program) (No. 2007CB814904). |
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Keywords: | Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 93E05 90C39 60H10 |
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