The Continuous Time Nonzero-sum Dynkin Game Problem and Application in Game Options

arXiv:0810.5698v1 [q-fin.PR] 31 Oct 2008 The Continuous Time Nonzero-sum Dynkin Game Problem and Application in Game Options Said Hamad`ene∗and Jianfeng Zhang† September 10, 2018 Abstract In this paper we study the nonzero-sum Dynkin game in continuous time which is a two player non-cooperative game on stopping times. We show that it has a Nash equilibrium point for general stochastic processes. As an application, we consider the problem of pricing American game contingent claims by the utility maximization approach. AMS Classification subjects: 91A15; 91A10; 91A30; 60G40; 91A60. Keywords: Nonzero-sum Game; Dynkin game; Snell envelope; Stopping time; Util- ity maximization; American game contingent claim. ∗Universit´e du Maine, D´epartement de Math´ematiques, Equipe Statistique et Processus, Avenue Olivier Messiaen, 72085 Le Mans, Cedex 9, France. e-mail: hamadene@univ-lemans.fr †USC Department of Mathematics, 3620 S. Vermont Ave, KAP 108, Los Angeles, CA 90089, USA. e-mail:jianfenz@usc.edu. Research supported in part by NSF grants DMS 04-03575 and DMS 06-31366. Part of the work was done while this author was visiting Universit´e du Maine, whose hospitality is greatly appreciated. 1 1 Introduction Dynkin games of zero-sum or nonzero-sum, continuous or discrete time types, are games on stopping times. Since their introduction by E.B. Dynkin in [10], they have attracted a lot of research activities (see e.g. [1, 2, 4, 5, 6, 7, 8, 11, 12, 14, 15, 19, 20, 21, 22, 23, 24, 25, 26] and the references therein). To begin with let us describe briefly those game problems. Assume we have a system controlled by two players or agents a1 and a2. The system works or is alive up to the time when one of the agents decides to stop the control at a stopping time τ1 for a1 and τ2 for a2. An example of that system is a recallable option in a financial market (see [15, 17] for more details). When the system is stopped the payment for a1 (resp. a2) amounts to a quantity J1(τ1, τ2) (resp. J2(τ1, τ2)) which could be negative and then it is a cost. We say that the nonzero-sum Dynkin game associated with J1 and J2 has a Nash equilibrium point (NEP for short) if there exists a pair of stopping times (τ ∗ 1 , τ ∗ 2) such that for any (τ1, τ2) we have: J1(τ ∗ 1 , τ ∗ 2 ) ≥J1(τ1, τ ∗ 2) and J2(τ ∗ 1 , τ ∗ 2) ≥J2(τ ∗ 1, τ2). The particular case where J1 + J2 = 0 corresponds to the zero-sum Dynkin game. In this case, when the pair (τ ∗ 1 , τ ∗ 2 ) exists it satisfies J1(τ ∗ 1 , τ2) ≤J1(τ ∗ 1 , τ ∗ 2) ≤J1(τ1, τ ∗ 2 ), for any τ1, τ2. We call such a (τ ∗ 1 , τ ∗ 2 ) a saddle-point for the game. Additionally this existence implies in particular that: inf τ1 sup τ2 J1(τ1, τ2) = sup τ2 inf τ1 J1(τ1, τ2), i.e., the game has a value. Mainly, in the zero-sum setting, authors aim at proving existence of the value or/and a saddle point for the game while in the nonzero-sum framework they focus on the issue of existence of a NEP for the game. In continuous time, for decades there have been a lot of works on zero-sum Dynkin games [1, 2, 5, 6, 8, 10, 11, 12, 15, 19, 20, 21, 25, 26]. Recently this type of game has attracted a new interest since it has been applied in mathematical finance (see 2 e.g. [3, 15, 16, 17]) in connection with the pricing of American game options intro- duced by Y.Kifer in [17]. Comparing with the zero-sum setting, there are much less results on nonzero-sum Dynkin games in the literature. Nevertheless in the Marko- vian framework, among other papers, one can quote [4, 7, 23, 24] which deal with the nonzero-sum Dynkin game. In non-Markovian framework, E.Etourneau [14] showed that the game has a NEP if some of the processes which define the game (Y 1 and Y 2 of (2.1) below) are supermartingales. Note that even in the Markovian setting, an equivalent condition is supposed. On the other hand, there are some other works which study the existence of approximate equilibrium points (see e.g. [21]). The main objective of this work is to study the existence of NEP for nonzero-sum Dynkin games in non-Markovian framework. For very general processes, we construct an NEP and thus it always exists. This removes the Etourneau’s type of conditions and, to our best knowledge, is novel in the literature. Our approach is based on the Snell envelope theory. We next apply our general existence result to price American Game Contingent Claim by the utility maximization approach. Kuhn [18] studied a similar problem by assuming that the agents a1 and a2 use only discrete stopping times and exponential utilities. We remove these constraints. The rest of the paper is organized as follows. In Section 2, we precise the setting of the problem and give some preliminary results related to the Snell envelope notion. In Section 3, we construct a sequence of pairs of decreasing stopping times and show that their limit pair is an NEP for the game. Finally in Section 4, we apply the result of Section 3 to price American Game Contingent Claim by the utility maximization a

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