Dynkin game under g-expectation in continuous time

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 V̲t=esssupτ∈Ttessinfσ∈TtEtg[R(τ,σ)]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\underline{V}_t=ess\sup \nolimits _{\tau \in {\mathcal {T}_t}} ess\inf \nolimits _{\sigma \in {\mathcal {T}_t}}\mathcal {E}^g_t[R(\tau ,\sigma )]$$\end{document} and V¯t=essinfσ∈Ttesssupτ∈TtEtg[R(τ,σ)]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{V}_t=ess\inf \nolimits _{\sigma \in {\mathcal {T}_t}} ess\sup \nolimits _{\tau \in {\mathcal {T}_t}}\mathcal {E}^g_t[R(\tau ,\sigma )]$$\end{document} are defined, respectively. Under some suitable assumptions, a pair of saddle points is obtained and the value function of Dynkin game V(t)=V̲t=V¯t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V(t)=\underline{V}_t=\overline{V}_t$$\end{document} follows. Furthermore, we also consider the constrained case of Dynkin game.

are defined, respectively, where R t (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 (τ * t , σ * t ), such that 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 (short for reflected BSDE) with one lower obstacle has been found very useful to solve this optimal stopping problem. Since then, many researchers have investigated the Dynkin game by solving the reflected BSDE with the lower and upper obstacles (for example, see Cvitanic and Karatzas [5], Hamadène and Lepeltier [11] and the references therein). In addition, there are some other ways to solve this game, such as the pathwise approach (see Karatzas [13]) and the connection with bounded variation problem (see Karatzas and Wang [14]).
Inspired by Cvitanic and Karatzas [5], in this paper we study Dynkin game in the stochastic environment with ambiguity and evaluate the reward process by g-expectations induced by BSDEs. Our problem can be formulated as follows. We define the lower and upper value functions and respectively, where R(τ, σ ) := L(τ )1 {τ ≤σ } + U (σ )1 {σ <τ } and T t is the set of all stopping times taking values in [t, T ]. Under some suitable assumptions on two processes L(t) and U (t), we want to find a pair of saddle points (τ * t , σ * t ) such that holds, and then the game has a value function This problem looks very similar to the one that was stated and solved in Cvitanic and Karatzas [5], but there are some differences between them. In Sect. 3, we point out the main difference between them. Furthermore, the more complicated case with a constraint is considered and the reward process is evaluated by g -expectation. The notion of g -expectation is given in Sect. 2.
This paper is organized as follows. In Sect. 2, we introduce some notations, assumptions, notions, propositions about BSDE, reflected BSDE and BSDE with a constraint that are used in this paper. The main results and proofs are stated in Sect. 3.

BSDE, reflected BSDE and BSDE with a constraint
In this section, we shall present some notations, assumptions, notions and propositions about BSDE, reflected BSDE and BSDE with a constraint that are used in this paper.
Let ( , F, P) be a probability space and (W t ) t≥0 be a d-dimensional standard Brownian motion with respect to filtration (F t ) t≥0 generated by Brownian motion and all P-null subsets, i.e., where N is the set of all P-null subsets. Fix a real number T > 0.
. Proposition 2.2 (see [23]) Suppose that g satisfies (A.0)-(A.2), and σ , τ are two stopping times satisfying τ ≤ σ ≤ T . Let ζ ∈ L 2 (F σ ) and (Y, Z ) be the solution of the following BSDE The theory of BSDE has been wildly used in many fields such as financial mathematics and stochastic control. Let us mention that the reflected BSDE has been found useful to solve the optimal stopping problem and investigate Dynkin game. In Cvitanic and Karatzas [5], Dynkin game is investigated by solving the reflected BSDE with double obstacles. In the following, the notion of reflected BSDE is given. [24]) Suppose that ξ ∈ L 2 (F T ), and g satisfies (A.0)-(A.2). Consider two processes L, U ∈ S 2 T (R) so as to satisfy
In this paper, the constrained case of Dynkin game is also considered. So we introduce the theory of BSDE with a constraint. First, the notion of g-supersolution is presented.
then we call Y a g-supersolution on [0, T ] Let φ : × [0, T ] × R × R d → R + be a given non-negative, F t -progressively measurable function such that φ(t, 0, 0) ∈ H 2 T (R) and φ is Lipschitz with respect to (y, z), that is, there exists a positive constant μ such that for all y 1 , y 2 ∈ R, z 1 , z 2 ∈ R d ,

Now, we consider BSDE of the form (11) with a constraint
imposed on the solution, that is, the solution of BSDE of the form (11) also satisfies that To make the problem of BSDE with a constraint meaningful, we need the following assumption: (H) There exists at least a g-supersolutionŶ and the corresponding decomposition (Ẑ ,Â) with terminal value ξ , such that the constraint (12) holds. Definition 2.7 (The smallest g-supersolution subject to a constraint, g -expectation, see [23]) Given terminal value ξ , a g-supersolution Y with the decomposition (Z , A) is said to be the smallest g-supersolution subject to the constraint (12), if for any g-supersolutionŶ with terminal value ξ and the corresponding decomposition (Ẑ ,Â) subject to φ(t,Ŷ t ,Ẑ t ) = 0 a.s., Y t ≤Ŷ t a.s. In the case of g(·, y, 0) = 0 and φ(·, y, 0) = 0, ∀y ∈ R, the smallest g-supersolution Y t subject to the constraint (12) Remark 2.8 To construct the smallest g-supersolution of BSDE (11) subject to the constraint (12), Peng [23] introduced the following BSDEs on [0, T ]: where g n := g + nφ. In [23], the author proved that Y n increasingly converges to Y , and Y is the smallest g-supersolution of BSDE (11) subject to the constraint (12).

Remark 2.9
Suppose that for any y ∈ R, g(·, y, 0) ≡ 0 and φ(·, y, 0) ≡ 0. Then the smallest g-supersolution subject to the constraint (12) is well defined for terminal value ξ in L ∞ (F T ), which denotes the space of all essentially bounded F T -measurable variables. In fact, for a given ξ ∈ L ∞ (F T ) (i.e., there exists a positive constant D such that |ξ | ≤ D a.s.), then the following Y with the corresponding decomposition is (Z , A) is a g-supersolution of BSDE (11) such that the constraint (12) holds, where and Z t = 0. Thus, the smallest g-supersolution subject to the constraint (12) exists.
At last, we give a comparison theorem of the smallest g-supersolution subject to the constraint (12).

Proposition 2.10 Suppose g and φ satisfy the assumptions (A.0), (A.2) and (A.3).
If ξ, η ∈ L 2 (F T ) and ξ ≤ η a.s., then Proof Let Y 1,n , Z 1,n and Y 2,n , Z 2,n be the solutions of the following BSDEs on [0, T ]: respectively. By the comparison theorem of BSDE (see Peng [22]), we can obtain that for each n, From Remark 2.8, we know the fact that lim The proof of Proposition 2.10 is complete.

Dynkin game under ambiguity
In this section, we first study the Dynkin game without constraints. Second, the more complicated case with a constraint will be investigated.

Dynkin game without constraints
In Cvitanic and Karatzas [5], the method for studying the Dynkin game is stated as follows. Define the lower and upper value functions respectively, where T t is the set of all stopping times taking values in [t, T ] and A pair of saddle points (τ * t , σ * t ) can be found, which satisfies and then the game has a value function In our framework, we need the assumption (A.3) to hold. The method for studying the Dynkin game in our framework can be formulated as follows. Define the lower and upper value functions and respectively, where R(τ, σ ) := L(τ )1 {τ ≤σ } + U (σ )1 {σ <τ } . By Definition 2.1, we know that where (Y τ,σ , Z τ,σ ) is the solution of BSDE (7) with terminal value η := L(τ )1 {τ ≤σ } + U (σ )1 {σ <τ } . We can find a pair of saddle points (τ * t , σ * t ) such that holds, and then the game has a value function V (t) = V t = V t . Furthermore, we can obtain that (16) also holds. From the expressions of (15) and (19), we can easily find the differences between our method and the method of Cvitanic and Karatzas [5]. In (15), (Y * , Z * ) is the solution of reflected BSDE (8) formulated in Definition 2.4 with terminal value L(T ), and do not depend on the stopping times σ and τ . But in (19), (Y τ,σ , Z τ,σ ) is the solution of BSDE (7) with terminal value L(τ )1 {τ ≤σ } + U (σ )1 {σ <τ } , and depend on the stopping times σ and τ .
With the help of the theory of g-expectation, we can find a pair of saddle points of the Dynkin game. The main reason is that g-expectation enjoys almost all properties of classical expectation except for linearity. But from the proof of Theorem 3.1, we can see that linearity is not crucial to study Dynkin game without constraints. Theorem 3.1 is the main result of the Dynkin game without constraints.  (17) and (18) has a pair of saddle points (τ * t , σ * t ), such that holds for any τ, σ ∈ T t , and hence and Proof For any τ ∈ T t , we have Now, we prove that for any τ ∈ T t , holds. Case 1: On the set {ω : σ * t < τ ≤ τ * t }, (24) obviously holds. Case 2: On the set {ω : τ ≤ τ * t < σ * t }, by (10) and (21), (22), we have A τ * t − A τ = 0 and K τ * t − K τ = 0. So BSDE (8) can be rewritten as follows: This means that Case 3: On the set {ω : τ ≤ σ * t ≤ τ * t }, by (10) and (21), (22), we have A σ * t − A τ = 0 and K σ * t − K τ = 0. So BSDE (8) can be rewritten as follows:

Dynkin game with a constraint
The Dynkin game with a constraint that we study can be formulated as follows. Define the lower and upper value functions