Probabilistic interpretation for systems of Isaacs equations with two reflecting barriers

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. Mathematics Subject Classification (2000). 93E05, 90C39, 60H10.

Mixed generalized Dynkin game and stochastic control in a Markovian framework

“…When the terminal reward map g is continuous, some optimization problems relative to BSDEs (see e.g. Peng [30]), to RBSDEs (see Buckdahn-Li [9]) or DRBSDEs (Buckdahn-Li [10] in the Brownian case) have been studied in the literature.…”

Some Recent Aspects of Differential Game Theory

“…In this section, based on the papers by Buckdahn and Li [35][36][37]39] the problem of existence of a value for zero-sum two-player stochastic differential games for a nonlinear payoff is revisited. This approach represent an alternative to that in the pioneering paper [65] on two-player zero-sum stochastic differential games stochastic differential games by Fleming and Souganidis.…”

“…On the other hand it seems also natural to consider nonlinear cost functionals and to allow the controls of the players to depend on events of the past which happened before the beginning of the game. The last two points have been investigated in a series of papers by Buckdahn and Li [35,36,39], and an approach more direct than that in [65] has been developed. The first point, together with the two others, will be the object of the fourth part of the survey.…”

Backward Stochastic Variational Inequalities with Locally Bounded Generators

“…The more general case of scalar BSDE with one-sided reflection and associated optimal control problems was considered by El Karoui et al [8] and with two-sided reflection associated with stochastic game problem by Cvitanic, Karatzas [6] (see also [3] and [7] for the investigation of zero-sum two-player stochastic differential games whose cost functionals are given by controlled reflected BSDE).…”