Dynkin game under g -expectation in continuous time
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Arabian Journal of Mathematics
Helin Wu · Yong Ren · Feng Hu
Dynkin game under g-expectation in continuous time
Received: 25 September 2019 / Accepted: 19 February 2020 © The Author(s) 2020
Abstract In this paper, we investigate some kind of Dynkin game under g-expectation induced by backward stochastic differential equation (short for BSDE). The lower and upper value functions V t = g g ess supτ ∈Tt ess inf σ ∈Tt Et [R(τ, σ )] and V t = ess inf σ ∈Tt ess supτ ∈Tt Et [R(τ, σ )] are defined, respectively. Under some suitable assumptions, a pair of saddle points is obtained and the value function of Dynkin game V (t) = V t = V t follows. Furthermore, we also consider the constrained case of Dynkin game. Mathematics Subject Classification
60H10
1 Introduction Dynkin game was first introduced and studied by Dynkin in [6]. Later, in Neveu [18], EIbakidze [7], Kifer [15] and Ohtsubo [19,20], the authors considered Dynkin game in discrete parameter case with (without) a finite constraint. The version of continuous time was also studied in many studies (for examples, see Morimoto [17], Stettner [26], Krylov [16] and the references therein). A general formulation of Dynkin game is stated as follows. The lower and upper value functions V t := ess sup ess inf E[Rt (τ, σ )|Ft ]
(1)
V t := ess inf ess sup E[Rt (τ, σ )|Ft ]
(2)
τ ∈ Tt
σ ∈ Tt
and σ ∈ Tt
τ ∈ Tt
H. Wu School of Mathematics and Statistics, Chongqing University of Technology, Chongqing 400054, China E-mail: [email protected] Y. Ren Department of Mathematics, Anhui Normal University, Wuhu 241000, China E-mail: [email protected] F. Hu (B) School of Statistics, Qufu Normal University, Qufu 273165, China E-mail: [email protected] F. Hu Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong, China
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are defined, respectively, where Rt (τ, σ ) is a function of two stopping times τ and σ . Then, one often tries to find a sufficient condition such that V t = V t holds. Obviously, V t ≥ V t . To get the reverse inequality, the method that we often use is to find a pair of saddle points (τt∗ , σt∗ ), such that E[Rt (τ, σt∗ )|Ft ] ≤ E[Rt (τt∗ , σt∗ )|Ft ] ≤ E[Rt (τt∗ , σ )|Ft ]
(3)
holds. If (3) is true, denote V (t) := V t = V t , and V (t) is called the value function of the Dynkin game. The Dynkin game can be seen as an extension of the optimal stopping problem. The martingale approach has been used to find a pair of saddle points, and then the value function is obtained by solving this double optimal stopping problem (for example, see Dynkin [6], Krylov [16] and the references therein). In Friedman [8] and Bensoussan and Friedman [1], the authors developed the analytical theory of stochastic differential games with stopping times in Markov setting. They studied the value function and found the saddle point of Dynkin game by using the theories of partial differential equations, variational inequalities and free-boundary problems. Later, reflected backward stochastic differential equation (shor
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