Stochastic Recursive Zero-Sum Differential Game and Mixed Zero-Sum Differential Game Problem

Abstract

Under the notable Issacs's condition on the Hamiltonian, the existence results of a saddle point are obtained for the stochastic recursive zero-sum differential game and mixed differential game problem, that is, the agents can also decide the optimal stopping time. The main tools are backward stochastic differential equations (BSDEs) and double-barrier reflected BSDEs. As the motivation and application background, when loan interest rate is higher than the deposit one, the American game option pricing problem can be formulated to stochastic recursive mixed zero-sum differential game problem. One example with explicit optimal solution of the saddle point is also given to illustrate the theoretical results.

1. Introduction

The nonlinear backward stochastic differential equations (BSDEs in short) had been introduced by Pardoux and Peng [1], who proved the existence and uniqueness of adapted solutions under suitable assumptions. Independently, Duffie and Epstein [2] introduced BSDE from economic background. In [2], they presented a stochastic differential recursive utility which is an extension of the standard additive utility with the instantaneous utility depending not only on the instantaneous consumption rate but also on the future utility. Actually, it corresponds to the solution of a particular BSDE whose generator does not depend on the variable 𝑍. From mathematical point of view, the result in [1] is more general. Then, El Karoui et al. [3] and Cvitanic and Karatzas [4] generalized, respectively, the results to BSDEs with reflection at one barrier and two barriers (upper and lower).

BSDE plays an important role in the theory of stochastic differential game. Under the notable Isaacs's condition, Hamadène and Lepeltier [5] obtained the existence result of a saddle point for zero-sum stochastic differential game with payoff 𝐽(𝑢,𝑣)=𝐸^{(𝑢,𝑣)𝑇}_𝑡𝑓𝑠,𝑥_𝑠,𝑢_𝑠,𝑣_𝑠𝑑𝑠+𝑔𝑥_𝑇. (1.1) Using a maximum principle approach, Wang and Yu [6, 7] proved the existence and uniqueness of an equilibrium point. We note that the cost function in [5] is not recursive, and the game system in [6, 7] is a BSDE. In [8], El Karoui et al. gave the formulation of recursive utilities and their properties from the BSDE's pointview. The problem that the cost function (payoff) of the game system is described by the solution of BSDE becomes the recursive differential game problem. In the following Section 2, we proved the existence of a saddle point for the stochastic recursive zero-sum differential game problem and also got the optimal payoff function by the solution of one specific BSDE. Here, the generator of the BSDE contains the main variable solution 𝑦_𝑡, and we extend the result in [5] to the recursive case which has much more significance in economics theory.

Then, in Section 3 we study the stochastic recursive mixed zero-sum differential game problem which is that the two agents have two actions, one is of control and the other is of stopping their strategies to maximize and minimize their payoffs. This kind of game problem without recursive variable and the American game option problem as this kind of mixed game problem can be seen in Hamadène [9]. Using the result of reflected BSDEs with two barriers, we got the saddle point and optimal stopping strategy for the recursive mixed game problem which has more general significance than that in [9].

In fact, the recursive (mixed) zero-sum game problem has wide application background in practice. When the loan interest rate is higher than the deposit one. The American game option pricing problem can be formulated to the stochastic recursive mixed game problem in our Section 3. To show the application of this kind of problem and our motivation to study our recursive (mixed) game problem, we analyze the American game option pricing problem and let it be an example in Section 4. We notice that in [5, 9], they did not give the explicit saddle point to the game, and it is very difficult for the general case. However, in Section 4, we also give another example of the recursive mixed zero-sum game problem, for which the explicit saddle point and optimal payoff function to illustrate the theoretical results.

2. Stochastic Recursive Zero-Sum Differential Game

In this section, we will study the existence of the stochastic recursive zero-sum differential game problem using the result of BSDEs.

Let {𝐵_𝑡,0≤𝑡≤𝑇} be an 𝑚-dimensional standard Brownian motion defined on a probability space (Ω,ℱ,𝑃). Let (ℱ_𝑡)_𝑡≥0 be the completed natural filtration of 𝐵_𝑡. Moreover,(i)𝒞is the space of continuous functions from [0,𝑇]to𝑅^𝑚;(ii)𝒫is the 𝜎-algebra on [0,𝑇]×Ωof ℱ_𝑡-progressively sets;(iii)for any stopping time𝜈,𝒯_𝜈 is the set of ℱ_𝑡-measurable stopping time𝜏 such that 𝑃-𝑎.𝑠.𝜈≤𝜏≤𝑇; 𝒯₀ will simply be denoted by𝒯;(iv)ℋ^2,𝑘is the set of 𝒫-measurable processes 𝜔=(𝜔_𝑡)_𝑡≤𝑇, 𝑅^𝑘-valued, and square integrable with respect to𝑑𝑡⊗𝑑𝒫;(v)𝒮²is the set of 𝒫-measurable and continuous processes𝜔^′=(𝜔^′_𝑡)_𝑡≤𝑇, such that 𝐸[sup_𝑡≤𝑇|𝜔^′_𝑡|²]<∞.

The 𝑚×𝑚 matrix 𝜎=(𝜎_𝑖𝑗) satisfies the following:(i)for any1≤𝑖, 𝑗≤𝑚,𝜎_𝑖𝑗is progressively measurable;(ii)for any(𝑡,𝑥)∈[0,𝑇]×𝒞, the matrix𝜎(𝑡,𝑥)is invertible;(iii)there exists a constants𝐾such that|𝜎(𝑡,𝑥)−𝜎(𝑡,𝑥^′)|≤𝐾|𝑥−𝑥^′|_𝑡 and |𝜎(𝑡,𝑥)|≤𝐾(1+|𝑥|_𝑡).

Then, the equation 𝑥_𝑡=𝑥₀+^𝑡₀𝜎𝑠,𝑥_𝑠𝑑𝐵_𝑠,𝑡≤𝑇 (2.1) has a unique solution (𝑥_𝑡).

Now, we consider a compact metric space 𝐴 (resp. 𝐵), and 𝒰 (resp. 𝒱) is the space of 𝒫-measurable processes 𝑢∶=(𝑢_𝑡)_𝑡≤𝑇 (resp. 𝑣∶=(𝑣_𝑡)_𝑡≤𝑇) with values in 𝐴 (resp. 𝐵). Let Φ∶[0,𝑇]×𝒞×𝒰×𝒱→𝑅^𝑚 be such that(i)for any (𝑡,𝑥)∈[0,𝑇]×𝒞, the mapping (𝑢,𝑣)→Φ(𝑡,𝑥,𝑢,𝑣)is continuous;(ii)for any(𝑢,𝑣)∈𝐴×𝐵, the functionΦ(⋅,𝑥(⋅),𝑢,𝑣)is 𝒫-measurable;(iii)there exists a constant𝐾such that|Φ(𝑡,𝑥,𝑢,𝑣)|≤𝐾(1+|𝑥|_𝑡)for any 𝑡, 𝑥, 𝑢, and𝑣;(iv)there exists a constant𝑀such that|𝜎⁻¹(𝑡,𝑥)Φ(𝑡,𝑥,𝑢,𝑣)|≤𝑀for any 𝑡, 𝑥, 𝑢, and𝑣.

For (𝑢,𝑣)∈𝒰×𝒱, we define the measure 𝑃^𝑢,𝑣 as 𝑑𝑃^𝑢,𝑣𝑑𝑃=exp^𝑇₀𝜎⁻¹𝑠,𝑥_𝑠Φ𝑠,𝑥_𝑠,𝑢_𝑠,𝑣_𝑠𝑑𝐵_𝑠−12^𝑇₀𝜎⁻¹𝑠,𝑥_𝑠Φ𝑠,𝑥_𝑠,𝑢_𝑠,𝑣_𝑠²𝑑𝑠. (2.2)

Thanks to Girsanov's theorem, under the probability 𝑃^𝑢,𝑣, the process 𝐵_𝑡^𝑢,𝑣=𝐵_𝑡−^𝑡₀𝜎⁻¹𝑠,𝑥_𝑠Φ𝑠,𝑥_𝑠,𝑢_𝑠,𝑣_𝑠𝑑𝑠,𝑡≤𝑇, (2.3) is a Brownian motion, and for this stochastic differential equation 𝑥_𝑡=𝑥₀+^𝑡₀Φ𝑠,𝑥_𝑠,𝑢_𝑠,𝑣_𝑠𝑑𝑠+^𝑡₀𝜎𝑠,𝑥_𝑠𝑑𝐵_𝑠^𝑢,𝑣,𝑡≤𝑇, (2.4)(𝑥_𝑡)_𝑡≤𝑇 is a weak solution.

Suppose that we have a system whose evolution is described by the process (𝑥_𝑡)_𝑡≤𝑇. On that system, two agents 𝑐₁ and 𝑐₂ intervene. A control action for 𝑐₁ (resp. 𝑐₂) is a process 𝑢=(𝑢_𝑡)_𝑡≤𝑇 (resp. 𝑣=(𝑣_𝑡)_𝑡≤𝑇) belonging to 𝒰 (resp. 𝒱). Thereby 𝒰 (resp. 𝒱) is called the set of admissible controls for 𝑐₁ (resp. 𝑐₂). When 𝑐₁ and 𝑐₂ act with, respectively, 𝑢 and 𝑣, the law of the dynamics of the system is the same as the one of 𝑥 under 𝑃^𝑢,𝑣. The two agents have no influence on the system, and they act to protect their advantages by means of 𝑢∈𝒰 and 𝑣∈𝒱 via the probability 𝑃^𝑢,𝑣.

In order to define the payoff, we introduce two functions 𝐶(𝑡,𝑥,𝑦,𝑢,𝑣) and 𝑔(𝑥) satisfying the following assumption: there exists 𝐿>0, for all 𝑥,𝑥^′∈ℋ^2,𝑚 and 𝑌,𝑌^′∈𝒮², such that 𝐶𝑡,𝑥_𝑡,𝑌_𝑡,𝑢,𝑣−𝐶𝑡,𝑥^′_𝑡,𝑌_𝑡,𝑢,𝑣≤𝐿𝑥_𝑡−𝑥^′_𝑡,𝑌_𝑡−𝑌^′_𝑡𝐶𝑡,𝑥_𝑡,𝑌_𝑡,𝑢,𝑣−𝐶𝑡,𝑥_𝑡,𝑌^′_𝑡,𝑢,𝑣≤𝐿𝑌_𝑡−𝑌^′_𝑡², (2.5) and 𝑔(𝑥) is measurable, Lipschitz continuous function with respect to 𝑥. The payoff 𝐽(𝑥₀,𝑢,𝑣) is given by 𝐽(𝑥₀,𝑢,𝑣)=𝑌₀, where 𝑌 satisfies the following BSDE: −𝑑𝑌_𝑠=𝐶𝑠,𝑥_𝑠,𝑌_𝑠,𝑢_𝑠,𝑣_𝑠𝑑𝑠−𝑍_𝑠𝑑𝐵_𝑠^𝑢,𝑣,𝑌_𝑇=𝑔𝑥_𝑇. (2.6) From the result in [10], there exists a unique solution (𝑌,𝑍) for 𝑢,𝑣. The agent 𝑐₁ wishes to minimize this payoff, and the agent 𝑐₂ wishes to maximize the same payoff. We investigate the existence of a saddle point for the game, more precisely a pair (𝑢^∗,𝑣^∗) of strategies, such that 𝐽(𝑥₀,𝑢^∗,𝑣)≤𝐽(𝑥₀,𝑢^∗,𝑣^∗)≤𝐽(𝑥₀,𝑢,𝑣^∗) for each (𝑢,𝑣)∈𝒰×𝒱.

For (𝑡,𝑥,𝑌,𝑍,𝑢,𝑣)∈[0,𝑇]×𝒞×𝑅×𝑅^𝑚×𝒰×𝒱, we introduce the Hamiltonian by 𝐻(𝑡,𝑥,𝑌,𝑍,𝑢,𝑣)=𝑍𝜎⁻¹(𝑡,𝑥)Φ(𝑡,𝑥,𝑢,𝑣)+𝐶(𝑡,𝑥,𝑌,𝑢,𝑣), (2.7) and we say that the Isaacs' condition holds if for (𝑡,𝑥,𝑌,𝑍)∈[0,𝑇]×𝒞×𝑅×𝑅^𝑚, max_𝑣∈𝒱min_𝑢∈𝒰𝐻(𝑡,𝑥,𝑌,𝑍,𝑢,𝑣)=min_𝑢∈𝒰max_𝑣∈𝒱𝐻(𝑡,𝑥,𝑌,𝑍,𝑢,𝑣). (2.8)

We suppose now that the Isaacs' condition is satisfied. By a selection theorem (see Benes [11]), there exists 𝑢^∗∶[0,𝑇]×𝒞×𝑅×𝑅^𝑚→𝒰, 𝑣^∗∶[0,𝑇]×𝒞×𝑅×𝑅^𝑚→𝒱, such that 𝐻𝑡,𝑥,𝑌,𝑍,𝑢^∗,𝑣≤𝐻𝑡,𝑥,𝑌,𝑍,𝑢^∗,𝑣^∗≤𝐻𝑡,𝑥,𝑌,𝑍,𝑢,𝑣^∗. (2.9)

Thanks to the assumption of 𝜎, Φ, and 𝐶, the function 𝐻(𝑡,𝑥,𝑌,𝑍,𝑢^∗(𝑡,𝑥,𝑌,𝑍),𝑣^∗(𝑡,𝑥,𝑌,𝑍)) is Lipschitz in 𝑍 and monotone in 𝑌 like the function 𝐶.

Now we give the main result of this section.

Theorem 2.1. (𝑌^∗,𝑍^∗) is the solution of the following BSDE: −𝑑𝑌^∗_𝑠=𝐻𝑠,𝑥_𝑠,𝑌^∗_𝑠,𝑍^∗_𝑠,𝑢^∗𝑠,𝑥,𝑌^∗,𝑍^∗,𝑣^∗𝑠,𝑥,𝑌^∗,𝑍^∗𝑑𝑠−𝑍^∗_𝑠𝑑𝐵_𝑠,𝑌^∗_𝑇=𝑔𝑥_𝑇. (2.10) Then, 𝑌^∗₀ is the optimal payoff 𝐽(𝑥₀,𝑢^∗,𝑣^∗), and the pair (𝑢^∗,𝑣^∗) is the saddle point for this recursive game.

Proof. We consider the following BSDE: 𝑌^∗_𝑡=𝑔𝑥_𝑇+^𝑇_𝑡𝐻𝑠,𝑥_𝑠,𝑌^∗_𝑠,𝑍^∗_𝑠,𝑢^∗𝑡,𝑥,𝑌^∗,𝑍^∗,𝑣^∗𝑡,𝑥,𝑌^∗,𝑍^∗𝑑𝑠−^𝑇_𝑡𝑍^∗_𝑠𝑑𝐵_𝑠. (2.11) Thanks to Theorem 2.1 in [10], the equation has a unique solution (𝑌^∗,𝑍^∗). Because 𝑌^∗₀ is deterministic, so 𝑌^∗₀=𝐸^{𝑢∗,𝑣∗}𝑌^∗₀=𝐸^{𝑢∗,𝑣∗}𝑔𝑥_𝑇+^𝑇₀𝐻𝑠,𝑥_𝑠,𝑌^∗_𝑠,𝑍^∗_𝑠,𝑢^∗𝑡,𝑥,𝑌^∗,𝑍^∗,𝑣^∗𝑡,𝑥,𝑌^∗,𝑍^∗𝑑𝑠−^𝑇₀𝑍^∗_𝑠𝑑𝐵_𝑠=𝐸^{𝑢∗,𝑣∗}𝑔𝑥_𝑇+^𝑇₀𝐶𝑠,𝑥_𝑠,𝑌^∗_𝑠,𝑢^∗_𝑠,𝑣^∗_𝑠𝑑𝑠−^𝑇₀𝑍^∗_𝑠𝑑𝐵^{𝑢∗,𝑣∗}_𝑠. (2.12) We can get 𝑌^∗₀=𝐽(𝑥₀,𝑢^∗,𝑣^∗).
For any 𝑢∈𝒰,𝑣∈𝒱, then we let 𝑌_𝑡=𝑔𝑥_𝑇+^𝑇_𝑡𝐶𝑠,𝑥_𝑠,𝑌_𝑠,𝑢^∗_𝑠,𝑣_𝑠𝑑𝑠−^𝑇_𝑡𝑍_𝑠𝑑𝐵^𝑢∗_𝑠^,𝑣=𝑔𝑥_𝑇+^𝑇_𝑡𝐻𝑠,𝑥_𝑠,𝑌_𝑠,𝑍_𝑠,𝑢^∗_𝑠,𝑣_𝑠𝑑𝑠−^𝑇_𝑡𝑍_𝑠𝑑𝐵_𝑠,𝑌^′_𝑡=𝑔𝑥_𝑇+^𝑇_𝑡𝐶𝑠,𝑥_𝑠,𝑌^′_𝑠,𝑢_𝑠,𝑣^∗_𝑠𝑑𝑠−^𝑇_𝑡𝑍^′_𝑠𝑑𝐵^𝑢,𝑣∗_𝑠=𝑔𝑥_𝑇+^𝑇_𝑡𝐻𝑠,𝑥_𝑠,𝑌^′_𝑠,𝑍^′_𝑠,𝑢_𝑠,𝑣^∗_𝑠𝑑𝑠−^𝑇_𝑡𝑍^′_𝑠𝑑𝐵_𝑠. (2.13) By the comparison theorem of the BSDEs and the inequality (2.9), we can compare the solutions of (2.11), and (2.13) and get 𝑌_𝑡≤𝑌^∗_𝑡≤𝑌^′_𝑡,  0≤𝑡≤𝑇,  so 𝑌₀=𝐽(𝑥₀,𝑢^∗,𝑣)≤𝐽(𝑥₀,𝑢^∗,𝑣^∗)≤𝐽(𝑥₀,𝑢,𝑣^∗)=𝑌^′₀ and (𝑢^∗,𝑣^∗) is the saddle point.

3. Stochastic Recursive Mixed Zero-Sum Differential Game

Now, we study the stochastic recursive mixed zero-sum differential game problem. First, let us briefly describe the problem.

Suppose now that we have a system, whose evolution also is described by (𝑥_𝑡)_{0≤𝑡≤𝑇}, which has an effect on the wealth of two controllers 𝐶₁ and 𝐶₂. On the other hand, the controllers have no influence on the system, and they act so as to protect their advantages, which are antagonistic, by means of 𝑢∈𝒰 for 𝐶₁ and 𝑣∈𝒱 for 𝐶₂ via the probability 𝑃^𝑢,𝑣 in (2.2). The couple (𝑢,𝑣)∈𝒰×𝒱 is called an admissible control for the game. Both controllers also have the possibility to stop controlling at 𝜏 for 𝐶₁ and 𝜃 for 𝐶₂; 𝜏 and 𝜃 are elements of 𝒯 which is the class of all ℱ_𝑡-stopping time. In such a case, the game stops. The controlling action is not free, and it corresponds to the actions of 𝐶₁ and 𝐶₂. A payoff is described by the following BSDE: 𝑌_𝑡^{𝑢,𝜏;𝑣,𝜃}=𝑈_𝜏1_[𝜏<𝜃]+𝐿_𝜃1_{[𝜃<𝜏<𝑇]}+𝑄_𝜏1_{[𝜏=𝜃<𝑇]}+𝑔𝑥_𝑇1_{[𝜏=𝜃=𝑇]}+_𝑡^𝜏∧𝜃𝐶𝑠,𝑥_𝑠,𝑌_𝑠^{𝑢,𝜏;𝑣,𝜃},𝑢_𝑠,𝑣_𝑠𝑑𝑠−_𝑡^𝜏∧𝜃𝑍_𝑠𝑑𝐵_𝑠^𝑢,𝑣, (3.1) and the payoff is given by 𝐽𝑥₀;𝑢,𝜏;𝑣,𝜃=𝑌₀^{𝑢,𝜏;𝑣,𝜃}=𝐸^(𝑢,𝑣)₀^𝜏∧𝜃𝐶𝑠,𝑥_𝑠,𝑌_𝑠^{𝑢,𝜏;𝑣,𝜃},𝑢_𝑠,𝑣_𝑠𝑑𝑠+𝑈_𝜏1_[𝜏<𝜃]+𝐿_𝜃1_{[𝜃<𝜏<𝑇]}+𝑄_𝜏1_{[𝜏=𝜃<𝑇]}+𝑔𝑥_𝑇1_{[𝜏=𝜃=𝑇]}, (3.2) where the (𝑈_𝑡)_𝑡≤𝑇, (𝐿_𝑡)_𝑡≤𝑇, and (𝑄_𝑡)_𝑡≤𝑇 are processes of 𝒮² such that 𝐿_𝑡≤𝑄_𝑡≤𝑈_𝑡. The action of 𝐶₁ is to minimize the payoff, and the action of 𝐶₂ is to maximize the payoff. Their terms can be understood as(i)𝐶(𝑠,𝑥,𝑌,𝑢,𝑣)is the instantaneous reward for𝐶₁and cost for𝐶₂;(ii)𝑈_𝜏is the cost for𝐶₁and for𝐶₂if𝐶₁decides to stop first the game;(iii)𝐿_𝜃is the reward for𝐶₂ and cost for 𝐶₁if𝐶₂decides stop first the game.

The problem is to find a saddle point strategy (one should say a fair strategy) for the controllers, that is, a strategy (𝑢^∗,𝜏^∗;𝑣^∗,𝜃^∗) such that 𝐽𝑥₀;𝑢^∗,𝜏^∗;𝑣,𝜃≤𝐽𝑥₀;𝑢^∗,𝜏^∗;𝑣^∗,𝜃^∗≤𝐽𝑥₀;𝑢,𝜏;𝑣^∗,𝜃^∗, (3.3) for any (𝑢,𝜏;𝑣,𝜃)∈𝒰×𝒯×𝒱×𝒯.

Like in Section 2, we also define the Hamiltonian associated with this mixed stochastic game problem by 𝐻(𝑡,𝑥,𝑌,𝑍,𝑢,𝑣), and thanks to the Benes's solution [11], there exist 𝑢^∗(𝑡,𝑥,𝑌,𝑍) and 𝑣^∗(𝑡,𝑥,𝑌,𝑍) satisfying 𝐻𝑡,𝑥,𝑌,𝑍,𝑢^∗(𝑡,𝑥,𝑌,𝑍),𝑣^∗(𝑡,𝑥,𝑌,𝑍)=max_𝑣∈𝒱min_𝑢∈𝒰𝑍𝜎⁻¹(𝑡,𝑥)Φ(𝑡,𝑥,𝑢,𝑣)+𝐶(𝑡,𝑥,𝑌,𝑢,𝑣)=min_𝑢∈𝒰max_𝑣∈𝒱𝑍𝜎⁻¹(𝑡,𝑥)Φ(𝑡,𝑥,𝑢,𝑣)+𝐶(𝑡,𝑥,𝑌,𝑢,𝑣). (3.4) It is easy to know that 𝐻(𝑡,𝑥,𝑌,𝑍,𝑢,𝑣) is Lipschitz in 𝑍 and monotone in 𝑌.

From the result in [12], the stochastic mixed zero-sum differential game problem is possibly connected with BSDEs with two reflecting barriers. Now, we give the main result of this section.

Theorem 3.1. (𝑌^∗,𝑍^∗,𝐾^∗+,𝐾^∗−) is the solution of the following reflected BSDE: 𝑌^∗_𝑡=𝑔𝑥_𝑇+^𝑇_𝑡𝐻𝑠,𝑥_𝑠,𝑌^∗_𝑠,𝑍^∗_𝑠,𝑢^∗_𝑠,𝑣^∗_𝑠𝑑𝑠+𝐾_𝑇^∗+−𝐾_𝑡^∗+−𝐾_𝑇^∗−−𝐾_𝑡^∗−−^𝑇_𝑡𝑍^∗_𝑠𝑑𝐵_𝑠, (3.5) satisfying forall0≤𝑡≤𝑇,𝐿_𝑡≤𝑌^∗_𝑡≤𝑈_𝑡, and ∫^𝑇₀(𝑌^∗_𝑠−𝐿_𝑠)𝑑𝐾_𝑠^∗+=∫^𝑇₀(𝑌^∗_𝑠−𝑈_𝑠)𝑑𝐾_𝑠^∗−=0.
One defines 𝜏^∗=inf{𝑠∈[0,𝑇],𝑌^∗_𝑠=𝑈_𝑠} and 𝜃^∗=inf{𝑠∈[0,𝑇],𝑌^∗_𝑠=𝐿_𝑠}.
Then 𝑌^∗₀=𝐽(𝑥₀;𝑢^∗,𝜏^∗;𝑣^∗,𝜃^∗), (𝑢^∗,𝜏^∗;𝑣^∗,𝜃^∗) is the saddle point strategy.

Proof. It is easy to know that the reflected BSDE (3.5) has a unique solution (𝑌^∗,𝑍^∗,𝐾^∗+,𝐾^∗−), then we have 𝑌^∗₀=𝑔𝑥_𝑇+^𝑇₀𝐻𝑠,𝑥_𝑠,𝑌^∗_𝑠,𝑍^∗_𝑠,𝑢^∗_𝑠,𝑣^∗_𝑠𝑑𝑠+𝐾^∗+−𝐾_𝑇^∗−−^𝑇₀𝑍^∗_𝑠𝑑𝐵_𝑠=𝑌^∗_{𝜏∗∧𝜃}^∗+^{𝜏∗∧𝜃∗}₀𝐶𝑠,𝑥_𝑠,𝑌^∗_𝑠,𝑢^∗_𝑠,𝑣^∗_𝑠𝑑𝑠+𝐾_𝜏^∗+_∗∧𝜃^∗−𝐾_𝜏^∗−_∗∧𝜃^∗−^{𝜏∗∧𝜃∗}₀𝑍^∗_𝑠𝑑𝐵^{𝑢∗,𝑣∗}_𝑠. (3.6) Since 𝐾^∗+ and 𝐾^∗− increase only when 𝑌 reaches 𝐿 and 𝑈, we have 𝐾_𝜏^∗+_∗∧𝜃^∗=𝐾_𝜏^∗−_∗∧𝜃^∗=0. As (∫^𝑡₀𝑍_𝑟𝑑𝐵^{𝑢∗,𝑣∗}_𝑟)_𝑡≤𝑇 is an (ℱ_𝑡,𝑃^{𝑢∗,𝑣∗})-martingale, then we get 𝑌^∗₀=𝐸^{𝑢∗,𝑣∗}𝑌^∗_{𝜏∗∧𝜃}^∗+^{𝜏∗∧𝜃∗}₀𝐶𝑠,𝑥_𝑠,𝑌^∗_𝑠,𝑢^∗_𝑠,𝑣^∗_𝑠𝑑𝑠+𝐾_𝜏^∗+_∗∧𝜃^∗−𝐾_𝜏^∗−_∗∧𝜃^∗−^{𝜏∗∧𝜃∗}₀𝑍^∗_𝑠𝑑𝐵^{𝑢∗,𝑣∗}_𝑠=𝐸^{𝑢∗,𝑣∗}𝑌^∗_{𝜏∗∧𝜃}^∗+^{𝜏∗∧𝜃∗}₀𝐶𝑠,𝑥_𝑠,𝑌^∗_𝑠,𝑢^∗_𝑠,𝑣^∗_𝑠𝑑𝑠. (3.7)
We know that 𝑌^∗_{𝜏∗∧𝜃}^∗=𝑌^∗_𝜏∗1_{[𝜏∗<𝜃}^∗_]+𝑌^∗_𝜃^∗1_[𝜃^∗_<𝜏∗]+𝑌^∗_𝜃^∗1_[𝜃^∗_{=𝜏∗<𝑇]}+𝑔(𝑥_𝑇)1_[𝜃^∗_{=𝜏∗=𝑇]} and 𝑌^∗_𝜏∗1_{[𝜏∗<𝜃}^∗_]=𝑈_𝜏∗1_{[𝜏∗<𝜃}^∗_], 𝑌^∗_𝜃^∗1_[𝜃^∗_<𝜏∗]=𝐿_𝜃^∗1_[𝜃^∗_<𝜏∗], 𝑌^∗_𝜃^∗1_[𝜃^∗_{=𝜏∗<𝑇]}=𝑄_𝜃^∗1_[𝜃^∗_{=𝜏∗<𝑇]}. So, 𝑌^∗₀=𝐸^{𝑢∗,𝑣∗}𝑈_𝜏∗1_[𝜃^∗_<𝜏∗]+𝐿_𝜃^∗1_[𝜃^∗_<𝜏∗]+𝑄_𝜃^∗1_[𝜃^∗_{=𝜏∗<𝑇]}+𝑔𝑥_𝑇1_[𝜃^∗_{=𝜏∗=𝑇]}+^{𝜏∗∧𝜃∗}₀𝐶𝑠,𝑥_𝑠,𝑌^∗_𝑠,𝑢^∗_𝑠,𝑣^∗_𝑠𝑑𝑠=𝐽𝑥₀,𝑢^∗,𝜏^∗;𝑣^∗,𝜃^∗. (3.8)
Next, let 𝑣_𝑡 be an admissible control, and let 𝜃∈𝒯. We desire to show that 𝑌^∗₀≥𝐽(𝑥₀,𝑢^∗,𝜏^∗;𝑣,𝜃). We have 𝑌^∗₀=𝑌^∗_{𝜏∗∧𝜃}+^𝜏∗₀^∧𝜃𝐻𝑠,𝑥_𝑠,𝑌^∗_𝑠,𝑍^∗_𝑠,𝑢^∗_𝑠,𝑣^∗_𝑠𝑑𝑠+𝐾_𝜏^∗+_∗∧𝜃−^𝜏∗₀^∧𝜃𝑍^∗_𝑠𝑑𝐵_𝑠=𝑈_𝜏∗1_{[𝜏∗<𝜃]}+𝑌^∗_𝜃1_{[𝜃<𝜏∗]}+𝑄_𝜃1_{[𝜃=𝜏∗<𝑇]}+𝑔𝑥_𝑇1_{[𝜃=𝜏∗=𝑇]}+^𝜏∗₀^∧𝜃𝐻𝑠,𝑥_𝑠,𝑌^∗_𝑠,𝑍^∗_𝑠,𝑢^∗_𝑠,𝑣^∗_𝑠𝑑𝑠+𝐾_𝜏^∗+_∗∧𝜃−^𝜏∗₀^∧𝜃𝑍^∗_𝑠𝑑𝐵_𝑠. (3.9)
The payoff 𝐽(𝑥₀,𝑢^∗,𝜏^∗;𝑣,𝜃) can be described by the solution of following BSDE: 𝑌₀=𝑈_𝜏∗1_{[𝜏∗<𝜃]}+𝐿_𝜃1_{[𝜃<𝜏∗<𝑇]}+𝑄_𝜃1_{[𝜏∗=𝜃<𝑇]}+𝑔𝑥_𝑇1_{[𝜏∗=𝜃=𝑇]}+^𝜏∗₀^∧𝜃𝐶𝑠,𝑥_𝑠,𝑌_𝑠,𝑢^∗_𝑠,𝑣_𝑠𝑑𝑠−^𝜏∗₀^∧𝜃𝑍_𝑠𝑑𝐵^𝑢∗_𝑠^,𝑣=𝑈_𝜏∗1_{[𝜏∗<𝜃]}+𝐿_𝜃1_{[𝜃<𝜏∗<𝑇]}+𝑄_𝜃1_{[𝜏∗=𝜃<𝑇]}+𝑔𝑥_𝑇1_{[𝜏∗=𝜃=𝑇]}+^𝜏∗₀^∧𝜃𝐻𝑠,𝑥_𝑠,𝑌_𝑠,𝑍_𝑠,𝑢^∗_𝑠,𝑣_𝑠𝑑𝑠−^𝜏∗₀^∧𝜃𝑍_𝑠𝑑𝐵_𝑠, (3.10) then 𝑌₀=𝐸^𝑢∗,𝑣𝑈_𝜏∗1_{[𝜏∗<𝜃]}+𝐿_𝜃1_{[𝜃<𝜏∗<𝑇]}+𝑄_𝜃1_{[𝜏∗=𝜃<𝑇]}+𝑔𝑥_𝑇1_{[𝜏∗=𝜃=𝑇]}+^𝜏∗₀^∧𝜃𝐻𝑠,𝑥_𝑠,𝑌_𝑠,𝑍_𝑠,𝑢^∗_𝑠,𝑣_𝑠𝑑𝑠−^𝜏∗₀^∧𝜃𝑍_𝑠𝑑𝐵_𝑠=𝐸^𝑢∗,𝑣𝑈_𝜏∗1_{[𝜏∗<𝜃]}+𝐿_𝜃1_{[𝜃<𝜏∗<𝑇]}+𝑄_𝜃1_{[𝜏∗=𝜃<𝑇]}+𝑔𝑥_𝑇1_{[𝜏∗=𝜃=𝑇]}+^𝜏∗₀^∧𝜃𝐶𝑠,𝑥_𝑠,𝑌_𝑠,𝑢^∗_𝑠,𝑣_𝑠𝑑𝑠, (3.11) and 𝐽(𝑥₀;𝑢^∗,𝜏^∗;𝑣,𝜃)=𝑌₀. Thanks to 𝐻(𝑠,𝑥_𝑠,𝑌_𝑠,𝑍_𝑠,𝑢^∗_𝑠,𝑣^∗_𝑠)≥𝐻(𝑠,𝑥_𝑠,𝑌_𝑠,𝑍_𝑠,𝑢^∗_𝑠,𝑣_𝑠), 𝑌^∗_𝜃1_{[𝜃<𝜏∗]}≥𝐿_𝜃1_{[𝜃<𝜏∗<𝑇]}, and 𝐾_𝜏^∗+_∗∧𝜃≥0 by the comparison theorem of BSDEs to compare (3.9) and (3.10) to get 𝑌^∗₀≥𝑌₀=𝐽(𝑥₀;𝑢^∗,𝜏^∗;𝑣,𝜃).
In the same way, we can show that 𝑌^∗₀=𝐽(𝑥₀;𝑢^∗,𝜏^∗;𝑣^∗,𝜃^∗)≤𝐽(𝑥₀;𝑢,𝜏;𝑣^∗,𝜃^∗) for any 𝜏∈𝒯 and any admissible control 𝑢. It follows that (𝑢^∗,𝜏^∗;𝑣^∗,𝜃^∗) is a saddle point for the recursive game.
Finally, let us show that the value of the game is 𝑌^∗₀. We have proved that 𝐽𝑥₀;𝑢^∗,𝜏^∗;𝑣,𝜃≤𝑌^∗₀=𝐽𝑥₀;𝑢^∗,𝜏^∗;𝑣^∗,𝜃^∗≤𝐽𝑥₀;𝑢,𝜏;𝑣^∗,𝜃^∗, (3.12) for any (𝑢,𝑣)∈𝒰×𝒱 and 𝜏,𝜃∈𝒯. Thereby, 𝑌^∗₀≤inf_{𝑢∈𝒰,𝜏∈𝒯}𝐽𝑥₀;𝑢,𝜏;𝑣^∗,𝜃^∗≤sup_{𝑣∈𝒱,𝜃∈𝒯}inf_{𝑢∈𝒰,𝜏∈𝒯}𝐽𝑥₀;𝑢,𝜏;𝑣,𝜃. (3.13) On the other hand, 𝑌^∗₀≥sup_{𝑣∈𝒱,𝜃∈𝒯}𝐽𝑥₀;𝑢^∗,𝜏^∗;𝑣,𝜃≥inf_{𝑢∈𝒰,𝜏∈𝒯}sup_{𝑣∈𝒱,𝜃∈𝒯}𝐽𝑥₀;𝑢,𝜏;𝑣,𝜃. (3.14) Now, due to the inequality inf_{𝑢∈𝒰,𝜏∈𝒯}sup_{𝑣∈𝒱,𝜃∈𝒯}𝐽𝑥₀;𝑢,𝜏;𝑣,𝜃≥sup_{𝑣∈𝒱,𝜃∈𝒯}inf_{𝑢∈𝒰,𝜏∈𝒯}𝐽𝑥₀;𝑢,𝜏;𝑣,𝜃, (3.15) we have 𝑌^∗₀=inf_{𝑢∈𝒰,𝜏∈𝒯}sup_{𝑣∈𝒱,𝜃∈𝒯}𝐽𝑥₀;𝑢,𝜏;𝑣,𝜃=sup_{𝑣∈𝒱,𝜃∈𝒯}inf_{𝑢∈𝒰,𝜏∈𝒯}𝐽𝑥₀;𝑢,𝜏;𝑣,𝜃. (3.16) The proof is now completed.

4. Application

In this section, we present two examples to show the applications of Section 3.

The first example is about the American game option pricing problem. We formulate it to be one stochastic recursive mixed game problem. This can be regarded as the application background of our stochastic game problem.

Example 4.1. American game option when loan interest is higher than deposit interest is shown.
In El Karoui et al. [13], they proved that the price of an American option corresponds to the solution of a reflected BSDE. And Hamadène [9] proved that the price of American game option corresponds to the solution of a reflected BSDE with two barriers. Now, we will show that under some constraints in financial market such as when loan interest rate is higher than deposit interest rate, the price of an American game option corresponds to the value function of stochastic recursive mixed zero-sum differential game problem.
We suppose that the investor is allowed to borrow money at time 𝑡 at an interest rate 𝑅_𝑡>𝑟_𝑡, where 𝑟_𝑡 is the bond rate. Then, the wealth of the investor satisfies −𝑑𝑋_𝑡=𝑏𝑡,𝑋_𝑡,𝑍_𝑡𝑑𝑡−𝑑𝐶_𝑡−𝑍_𝑡𝑑𝑊_𝑡𝑏,0≤𝑡≤𝑇,𝑡,𝑋_𝑡,𝑍_𝑡∶=−𝑟_𝑡𝑋_𝑡+𝜃_𝑡𝑍_𝑡−𝑅_𝑡−𝑟_𝑡𝑋_𝑡−𝑍_𝑡𝜎_𝑡⁻, (4.1) where 𝑍_𝑡∶=𝜎_𝑡𝜋_𝑡,𝜃_𝑡∶=𝜎_𝑡⁻¹(𝑏_𝑡−𝑟_𝑡). 𝑏_𝑡 represents the instantaneous expected return rate in stock, 𝜎_𝑡 which is invertible represents the instantaneous volatility of the stock, and 𝐶_𝑡 is interpreted as a cumulative consumption process. 𝑏_𝑡, 𝑟_𝑡, 𝑅_𝑡, and 𝜎_𝑡 are all deterministic bounded functions, and 𝜎_𝑡⁻¹ is also bounded.
An American game is a contract between a broker 𝑐₁ and a trader 𝑐₂ who are, respectively, the seller and the buyer of the option. The trader pays an initial amount (the price of the option) which guarantees a payment of (𝐿_𝑡)_𝑡≤𝑇. The trader can exercise whenever he decides before the maturity 𝑇 of the option. Thus, if the trader decides to exercise at 𝜃, he gets the amount 𝐿_𝜃. On the other hand, the broker is allowed to cancel the contract. Therefore, if he chooses 𝜏 as the contract cancellation time, he pays the amount 𝑈_𝜏, and 𝑈_𝜏≥𝐿_𝜏. The difference 𝑈_𝜏−𝐿_𝜏 is the premium that the broker pays for his decision to cancel the contract. If 𝑐₁ and 𝑐₂ decide together to stop the contract at the time 𝜏, then 𝑐₂ gets a reward equal to 𝑄_𝜏1_[𝜏<𝑇]+𝜉1_[𝜏=𝑇]. Naturally, 𝑈_𝜏≥𝑄_𝜏≥𝐿_𝜏. 𝑈_𝑡, 𝐿_𝑡, and 𝑄_𝑡 are stochastic processes which are related to the stock price in the market.
We consider the problem of pricing an American game contingent claim at each time 𝑡 which consists of the selection of a stopping time 𝜏∈ℱ_𝜏 (or 𝜃∈ℱ_𝜃) and a payoff 𝑈_𝜏 (or 𝐿_𝜃) on exercise if 𝜏<𝜃<𝑇 (or 𝜃<𝜏<𝑇) and 𝜉 if 𝜏=𝑇. Set 𝑆_𝜏∧𝜃=𝜉1_{{𝜏=𝜃=𝑇}}+𝑄_𝜏1_{{𝜏=𝜃<𝑇}}+𝐿_𝜃1_{{𝜃<𝜏<𝑇}}+𝑈_𝜏1_{{𝜏<𝜃<𝑇}},0≤(𝜏∧𝜃)≤𝑇, (4.2) then the price of American game contingent claim (𝑆_𝜏∧𝜃,0≤(𝜏∧𝜃)≤𝑇) at time 𝑡 is given by 𝑋_𝑡=essinf_{𝜏∈ℱ𝜏}esssup_{𝜃∈ℱ𝜃}𝑋_𝑡𝜏∧𝜃,𝑆_𝜏∧𝜃, (4.3) where 𝑋_𝑡(𝜏∧𝜃,𝑆_𝜏∧𝜃) noted by 𝑋_𝑡^𝜏∧𝜃 satisfies BSDE −𝑑𝑋_𝑠^𝜏∧𝜃=𝑏𝑠,𝑋_𝑠^𝜏∧𝜃,𝑍_𝑠^𝜏∧𝜃𝑑𝑠−𝑑𝐶_𝑠−𝑍_𝑠^𝜏∧𝜃𝑑𝑊_𝑠,𝑋^𝜏∧𝜃_𝜏∧𝜃=𝑆_𝜏∧𝜃. (4.4) For each (𝜔,𝑡), 𝑏(𝑡,𝑥,𝑧) is a convex function of (𝑥,𝑧). It follows from [14] that we have 𝑋_𝑡^𝜏∧𝜃=esssup_{𝑟𝑡≤𝛽𝑡≤𝑅𝑡}essinf_𝐶𝑡𝑋^{𝛽,𝐶,𝜏∧𝜃}. Here, 𝑋^{𝛽,𝐶,𝜏∧𝜃} satisfies −𝑑𝑋_𝑠^{𝛽,𝐶,𝜏∧𝜃}=𝑏^𝛽𝑠,𝑋_𝑠^{𝛽,𝐶,𝜏∧𝜃},𝑍_𝑠^{𝛽,𝐶,𝜏∧𝜃}𝑑𝑠−𝑑𝐶_𝑠−𝑍_𝑠^𝜏∧𝜃𝑑𝑊_𝑠,𝑋^{𝛽,𝐶,𝜏∧𝜃}_𝜏∧𝜃=𝑆_𝜏∧𝜃,𝑏^𝛽𝑠,𝑋_𝑡,𝑍_𝑡∶=−𝛽_𝑡𝑋_𝑡−𝜃_𝑡+𝑟_𝑡−𝛽_𝑡𝜎_𝑡𝑍_𝑡, (4.5) where 𝛽_𝑡 is a bounded 𝑅-valued adapted process which can be regarded as an interest rate process in finance. So, 𝑋_𝑡∶=essinf_{𝜏∈ℱ𝜏}esssup_{𝜃∈ℱ𝜃}𝑋_𝑡𝜏∧𝜃,𝑆_𝜏∧𝜃=essinf_{𝜏∈ℱ𝑡,𝐶𝑡}esssup_{𝜃∈ℱ𝑡,𝑟𝑡≤𝛽𝑡≤𝑅𝑡}𝑋_𝑡^{𝛽,𝐶,𝜏∧𝜃}=esssup_{𝜃∈ℱ𝑡,𝑟𝑡≤𝛽𝑡≤𝑅𝑡}essinf_{𝜏∈ℱ𝑡,𝐶𝑡}𝑋_𝑡^{𝛽,𝐶,𝜏∧𝜃}. (4.6) Here, 𝑋_𝑡^𝛽,𝐶∶=esssup_{𝜃∈ℱ𝑡,𝑟𝑡≤𝛽𝑡≤𝑅𝑡}essinf_{𝜏∈ℱ𝑡,𝐶𝑡}𝑋_𝑡^{𝛽,𝐶,𝜏∧𝜃}. Then, from [13], there exist 𝑍_𝑡^𝛽,𝐶∈𝐻² and 𝐾_𝑡^𝛽,𝐶,+𝐾_𝑡^{𝛽,𝐶,−}, which are increasing adapted continuous processes with 𝐾₀^𝛽,𝐶,+=0 and 𝐾₀^{𝛽,𝐶,−}=0, such that (𝑋_𝑡^𝛽,𝐶,𝑍_𝑡^𝛽,𝐶,𝐾_𝑡^𝛽,𝐶,+,𝐾_𝑡^{𝛽,𝐶,−}) satisfies the following reflected BSDE: −𝑑𝑋_𝑠^𝛽,𝐶=𝑏^𝛽𝑠,𝑋_𝑠^𝛽,𝐶,𝑍_𝑠^𝛽,𝐶𝑑𝑠−𝑑𝐶_𝑠+𝑑𝐾_𝑠^𝛽,𝐶,+−𝑑𝐾_𝑠^{𝛽,𝐶,−}−𝑍_𝑠^𝛽,𝐶𝑑𝑊_𝑠,𝑋_𝑇^𝛽,𝐶=𝜉,0≤𝑠≤𝑇, (4.7) with 𝑈_𝑡≥𝑋_𝑡^𝛽,𝐶≥𝐿_𝑡, 0≤𝑡≤𝑇, and ∫^𝑇₀(𝑋_𝑡^𝛽,𝐶−𝐿_𝑡)⁻𝑑𝐾_𝑡^𝛽,𝐶,+=0, ∫^𝑇₀(𝑈_𝑡−𝑋_𝑡^𝛽,𝐶)⁻𝑑𝐾_𝑡^{𝛽,𝐶,−}=0. Then, the stopping time 𝜏=inf{𝑡≤𝑠≤𝑇;𝑋_𝑠^𝛽,𝐶=𝑈_𝑠}, and 𝜃=inf{𝑡≤𝑠≤𝑇;𝑋_𝑠^𝛽,𝐶=𝐿_𝑠}.
We formulate the pricing problem of American game option to the stochastic recursive mixed zero-sum differential game problem which was studied in Section 3, so the previous example provides the practical background for our problem. This is also one of our motivations to study the recursive mixed game problem in this paper.

In the following, we give another example, where we obtain the explicit saddle point strategy and optimal value of the stochastic recursive game. The purpose of this example is to illustrate the application of our theoretical results.

Example 4.2. We let the dynamics of the system (𝑥_𝑡)_𝑡≤𝑇 satisfy 𝑑𝑥_𝑡=𝑥_𝑡𝑑𝐵_𝑡,𝑡≤1,wheretheinitialvalueis𝑥₀. (4.8) The control action for 𝑐₁ (resp. 𝑐₂) is 𝑢 (resp. 𝑣) which belongs to 𝒰 (resp. 𝒱). The 𝒰 is [0,1], and the 𝒱 is [0,1], while the function Φ=𝑥_𝑡(𝑢_𝑡+𝑣_𝑡). Then, by the Girsanov's theorem, we can define the probability 𝑃^𝑢,𝑣 by 𝑑𝑃^𝑢,𝑣𝑑𝑃=exp^𝑇₀𝑢_𝑠+𝑣_𝑠𝑑𝐵_𝑠−12^𝑇₀𝑢_𝑠+𝑣_𝑠²𝑑𝑠. (4.9) Under the probability 𝑃^𝑢,𝑣, the process 𝐵_𝑡^𝑢,𝑣=𝐵_𝑡−∫^𝑡₀(𝑢_𝑠+𝑣_𝑠)𝑑𝑠 is a Brownian motion.
First, we consider the following stochastic recursive zero-sum differential game: 𝐽𝑥₀,𝑢,𝑣=𝑌₀=𝐸^𝑢,𝑣𝑥_𝑇+^𝑇₀min𝑥_𝑡,2+𝑌_𝑡𝑢_𝑡+𝑣_𝑡𝑑𝑡. (4.10)(𝑌_𝑡)_{0≤𝑡≤𝑇} satisfies BSDE −𝑑𝑌_𝑠=min𝑥_𝑠,2+𝑌_𝑠𝑢_𝑠+𝑣_𝑠𝑑𝑠−𝑍_𝑠𝑑𝐵_𝑠^𝑢,𝑣,𝑌_𝑇=𝑥_𝑇. (4.11) Therefore, 𝐻(𝑡,𝑥,𝑧,𝑌,𝑢,𝑣)=𝑍(𝑢+𝑣)+min𝑥_𝑡,2+𝑌(𝑢+𝑣), (4.12) and obviously, the Isaacs condition is satisfied with 𝑢^∗=1_{[𝑍+𝑌≤0]},𝑣^∗=1_{[𝑍+𝑌≥0]}. It follows that min_𝑢∈𝒰max_𝑣∈𝒱𝐻(𝑡,𝑥,𝑍,𝑌,𝑢,𝑣)=max_𝑣∈𝒱min_𝑢∈𝒰𝐻(𝑡,𝑥,𝑍,𝑌,𝑢,𝑣)=𝑍+min𝑥_𝑡𝐽𝑥,2+𝑌,₀,𝑢^∗,𝑣^∗=𝑌₀=𝑥_𝑇+^𝑇₀𝑍_𝑡+min𝑥_𝑡,2+𝑌_𝑡𝑑𝑡𝑣−^𝑇₀𝑍_𝑡𝑑𝐵_𝑡=𝐸𝑥₀exp2𝐵_𝑇+^𝑇₀exp𝐵_𝑡+12𝑡min𝑥₀exp𝐵_𝑡−12𝑡,2𝑑𝑡. (4.13) We also can get the conclusion that the optimal game value 𝑌^∗₀=𝐽(𝑥₀,𝑢^∗,𝑣^∗) is an increasing function with the initial value of the dynamics system 𝑥₀ from the previous representation. Now, we give the numerical simulation and draw Figure 1 to show this point. Let 𝑇=2, when 𝑥₀=1, the optimal game value 𝑌₀=147.8, 𝑍₀=147.8 and the saddle point strategy (𝑢^∗₀,𝑣^∗₀)=(0,1); when 𝑥₀=2, 𝑌₀=295.6, 𝑍₀=295.6, (𝑢^∗₀,𝑣^∗₀)=(0,1); and𝑥₀=3, 𝑌₀=443.4, 𝑍₀=443.4, and (𝑢^∗₀,𝑣^∗₀)=(0,1). 𝑌₀ is increasing function of 𝑥₀ which coincides with our conclusion.
Second, we consider the following stochastic recursive mixed zero-sum differential game: 𝐽𝑥₀;𝑢,𝜏;𝑣,𝜃=𝑌₀^{𝑢,𝜏;𝑣,𝜃}=𝐸^𝑢,𝑣₀^𝜏∧𝜃min𝑥_𝑡,2+𝑌_𝑡𝑢_𝑡+𝑣_𝑡+𝑥𝑑𝑡_𝜏+1𝐼_[𝜏<𝜃]+𝑥_𝜃−1𝐼_{[𝜃<𝜏<𝑇]}+𝑥_𝑇𝐼_[𝜃=𝜏]. (4.14) Then, (𝑌_𝑡)_{0≤𝑡≤(𝜏∧𝜃)} satisfies the following BSDE: 𝑌_𝑡=𝑥_𝜏+1𝐼_[𝜏<𝜃]+𝑥_𝜃−1𝐼_{[𝜃<𝜏<𝑇]}+𝑥_𝑇𝐼_[𝜃=𝜏]+_𝑡^𝜏∧𝜃min𝑥_𝑠,2+𝑌_𝑠𝑢_𝑠+𝑣_𝑠𝑑𝑠−_𝑡^𝜏∧𝜃𝑍_𝑠𝑑𝐵_𝑠^𝑢,𝑣. (4.15) Therefore, 𝐻(𝑡,𝑥,𝑧,𝑌,𝑢,𝑣)=𝑍(𝑢+𝑣)+min{|𝑥_𝑡|,2}+𝑌(𝑢+𝑣), and obviously, the Isaacs condition is satisfied with 𝑢^∗=1_{[𝑍+𝑌≤0]},𝑣^∗=1_{[𝑍+𝑌≥0]}. It follows that min_𝑢∈𝒰max_𝑣∈𝒱𝐻(𝑡,𝑥,𝑍,𝑌,𝑢,𝑣)=max_𝑣∈𝒱min_𝑢∈𝒰𝐻(𝑡,𝑥,𝑍,𝑌,𝑢,𝑣)=𝑍+min𝑥_𝑡𝐽𝑥,2+𝑌,₀;𝑢^∗,𝜏;𝑣^∗,𝜃=𝑌^{𝑢∗,𝜏;𝑣∗}₀^,𝜃=𝑌_𝜏∧𝜃+₀^𝜏∧𝜃𝑍_𝑡+min𝑥_𝑡,2+𝑌_𝑡𝑑𝑡−₀^𝜏∧𝜃𝑍_𝑡𝑑𝐵_𝑡=𝑌_𝜏∧𝜃1exp2(𝜏∧𝜃)+𝐵_𝜏∧𝜃+₀^𝜏∧𝜃min𝑥_𝑡1,2exp2(𝑡)+𝐵_𝑡−𝑑𝑡_0𝜏∧𝜃1exp2(𝑡)+𝐵_𝑡𝑍_𝑡+𝑌_𝑡𝑑𝐵_𝑡, (4.16) where 𝜏^∗=inf{𝑡∈[0,𝑇],𝑌_𝑡≥(𝑥_𝑡+1)}, and 𝜃^∗=inf{𝑡∈[0,𝑇],𝑌_𝑡≤(𝑥_𝑡−1)}, while (𝑢^∗,𝜏^∗;𝑣^∗,𝜃^∗) is the saddle point. So, the optimal value is 𝐽𝑥₀;𝑢^∗,𝜏^∗;𝑣^∗,𝜃^∗=𝑌^{𝑢∗,𝜏∗;𝑣∗,𝜃∗}₀=𝑌_{𝜏∗∧𝜃}^∗1exp2𝜏^∗∧𝜃^∗+𝐵_{𝜏∗∧𝜃}^∗+^{𝜏∗∧𝜃∗}₀min𝑥_𝑡1,2exp2𝑡+𝐵_𝑡−𝑑𝑡_{𝜏∗∧𝜃∗0}1exp2𝑡+𝐵_𝑡𝑍_𝑡+𝑌_𝑡𝑑𝐵_𝑡=𝐸𝑥₀exp2𝐵_{𝜏∗∧𝜃}^∗+1_𝜏∗<𝜃^∗exp𝐵^∗_𝜏+12𝜏^∗−1_𝜃^∗_<𝜏∗exp𝐵^∗_𝜃+12𝜃^∗+^{𝜏∗∧𝜃∗}₀min𝑥₀exp𝐵_𝑡−12𝑡1,2exp2𝑡+𝐵_𝑡𝑑𝑡. (4.17)
We also can get the conclusion that the optimal game value 𝑌^∗₀=𝐽(𝑥₀,𝑢^∗,𝜏^∗;𝑣^∗,𝜃^∗) is an increasing function with the initial value of the dynamics system 𝑥₀ from the previous representation.

Figure 1 

𝑌0 stands for the optimal game value, and 𝑥0 stand for the initial value of the dynamics system.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (no. 10921101, 61174092), the National Science Fund for Distinguished Young Scholars of China (no. 11125102), and the Special Research Foundation for Young teachers of Ocean University of China (no. 201313006).