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

Authors: Lifeng Wei; Zhen Wu
Journal: Mathematical Problems in Engineering
 (2012)

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## 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.


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## Body

## 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
(1.1)J(u,v)=E(u,v)[∫tT‍f(s,xs,us,vs)ds+g(xT)].
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 yt, and we extend the result in [5] to the recursive case which has much more significance in economics theory.Then, in Section3 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 Section3. 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{Bt,0≤t≤T} be an m-dimensional standard Brownian motion defined on a probability space (Ω,ℱ,P). Let (ℱt)t≥0 be the completed natural filtration of Bt. Moreover,(i)
𝒞is the space of continuous functions from [0,T]toRm;(ii)
𝒫is the σ-algebra on [0,T]×Ωof ℱt-progressively sets;(iii)
for any stopping timeν,𝒯ν is the set of ℱt-measurable stopping timeτ such that P-a.s.ν≤τ≤T; 𝒯0 will simply be denoted by𝒯;(iv)
ℋ2,kis the set of 𝒫-measurable processes ω=(ωt)t≤T, Rk-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[supt≤T|ωt′|2]<∞.Them×m matrix σ=(σij) satisfies the following:(i)
for any1≤i, j≤m,σijis 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(2.1)xt=x0+∫0t‍σ(s,xs)dBs,t≤T
has a unique solution (xt).Now, we consider a compact metric spaceA (resp. B), and 𝒰 (resp. 𝒱) is the space of 𝒫-measurable processes u:=(ut)t≤T (resp. v:=(vt)t≤T) with values in A (resp. B). Let Φ:[0,T]×𝒞×𝒰×𝒱→Rm 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 Pu,v as
(2.2)dPu,vdP=exp{∫0T‍σ-1(s,xs)Φ(s,xs,us,vs)dBs-12∫0T‍|σ-1(s,xs)Φ(s,xs,us,vs)|2ds}.Thanks to Girsanov's theorem, under the probabilityPu,v, the process
(2.3)Btu,v=Bt-∫0t‍σ-1(s,xs)Φ(s,xs,us,vs)ds,t≤T,
is a Brownian motion, and for this stochastic differential equation
(2.4)xt=x0+∫0t‍Φ(s,xs,us,vs)ds+∫0t‍σ(s,xs)dBsu,v,t≤T,(xt)t≤T is a weak solution.Suppose that we have a system whose evolution is described by the process(xt)t≤T. On that system, two agents c1 and c2 intervene. A control action for c1 (resp. c2) is a process u=(ut)t≤T (resp. v=(vt)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 Pu,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 Pu,v.In order to define the payoff, we introduce two functionsC(t,x,y,u,v) and g(x) satisfying the following assumption: there exists L>0, for all x,x'∈ℋ2,m and Y,Y'∈𝒮2, such that
(2.5)|C(t,xt,Yt,u,v)-C(t,xt′,Yt,u,v)|≤L|xt-xt′|,(Yt-Yt′)(C(t,xt,Yt,u,v)-C(t,xt,Yt′,u,v))≤L(Yt-Yt′)2,
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:
(2.6)-dYs=C(s,xs,Ys,us,vs)ds-ZsdBsu,v,YT=g(xT).
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×Rm×𝒰×𝒱, we introduce the Hamiltonian by
(2.7)H(t,x,Y,Z,u,v)=Zσ-1(t,x)Φ(t,x,u,v)+C(t,x,Y,u,v),
and we say that the Isaacs' condition holds if for (t,x,Y,Z)∈[0,T]×𝒞×R×Rm,
(2.8)maxv∈𝒱minu∈𝒰H(t,x,Y,Z,u,v)=minu∈𝒰maxv∈𝒱H(t,x,Y,Z,u,v).We suppose now that the Isaacs' condition is satisfied. By a selection theorem (see Benes [11]), there exists u*:[0,T]×𝒞×R×Rm→𝒰, v*:[0,T]×𝒞×R×Rm→𝒱, such that
(2.9)H(t,x,Y,Z,u*,v)≤H(t,x,Y,Z,u*,v*)≤H(t,x,Y,Z,u,v*).Thanks to the assumption ofσ, 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:
(2.10)-dYs*=H(s,xs,Ys*,Zs*,u*(s,x,Y*,Z*),v*(s,x,Y*,Z*))ds-Zs*dBs,YT*=g(xT).
Then, Y0* 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:(2.11)Yt*=g(xT)+∫tT‍H(s,xs,Ys*,Zs*,u*(t,x,Y*,Z*),v*(t,x,Y*,Z*))ds-∫tT‍Zs*dBs.
Thanks to Theorem 2.1 in [10], the equation has a unique solution (Y*,Z*). Because Y0* is deterministic, so
(2.12)Y0*=Eu*,v*[Y0*]=Eu*,v*[g(xT)+∫0T‍H(s,xs,Ys*,Zs*,u*(t,x,Y*,Z*),v*(t,x,Y*,Z*))ds-∫0T‍Zs*dBs]=Eu*,v*[g(xT)+∫0T‍C(s,xs,Ys*,us*,vs*)ds-∫0T‍Zs*dBsu*,v*].
We can get Y0*=J(x0,u*,v*).
For anyu∈𝒰,v∈𝒱, then we let
(2.13)Yt=g(xT)+∫tT‍C(s,xs,Ys,us*,vs)ds-∫tT‍ZsdBsu*,v=g(xT)+∫tT‍H(s,xs,Ys,Zs,us*,vs)ds-∫tT‍ZsdBs,Yt′=g(xT)+∫tT‍C(s,xs,Ys′,us,vs*)ds-∫tT‍Zs′dBsu,v*=g(xT)+∫tT‍H(s,xs,Ys′,Zs′,us,vs*)ds-∫tT‍Zs′dBs.
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 Yt≤Yt*≤Yt′,  0≤t≤T,  so Y0=J(x0,u*,v)≤J(x0,u*,v*)≤J(x0,u,v*)=Y0′ 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(xt)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 Pu,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 ℱt-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:
(3.1)Ytu,τ;v,θ=Uτ1[τ<θ]+Lθ1[θ<τ<T]+Qτ1[τ=θ<T]+g(xT)1[τ=θ=T]+∫tτ∧θ‍C(s,xs,Ysu,τ;v,θ,us,vs)ds-∫tτ∧θ‍ZsdBsu,v,
and the payoff is given by
(3.2)J(x0;u,τ;v,θ)=Y0u,τ;v,θ=E(u,v)[∫0τ∧θ‍C(s,xs,Ysu,τ;v,θ,us,vs)ds+Uτ1[τ<θ]+Lθ1[θ<τ<T]E(u,v)E+Qτ1[τ=θ<T]+g(xT)1[τ=θ=T]∫0τ∧θ‍],
where the (Ut)t≤T, (Lt)t≤T, and (Qt)t≤T are processes of 𝒮2 such that Lt≤Qt≤Ut. 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)
Uτis the cost forC1and forC2ifC1decides to stop first the game;(iii)
Lθis 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
(3.3)J(x0;u*,τ*;v,θ)≤J(x0;u*,τ*;v*,θ*)≤J(x0;u,τ;v*,θ*),
for any (u,τ;v,θ)∈𝒰×𝒯×𝒱×𝒯.Like in Section2, 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
(3.4)H(t,x,Y,Z,u*(t,x,Y,Z),v*(t,x,Y,Z))=maxv∈𝒱minu∈𝒰[Zσ-1(t,x)Φ(t,x,u,v)+C(t,x,Y,u,v)]=minu∈𝒰maxv∈𝒱[Zσ-1(t,x)Φ(t,x,u,v)+C(t,x,Y,u,v)].
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:
(3.5)Yt*=g(xT)+∫tT‍H(s,xs,Ys*,Zs*,us*,vs*)ds+(KT*+-Kt*+)-(KT*--Kt*-)-∫tT‍Zs*dBs,
satisfying forall0≤t≤T,Lt≤Yt*≤Ut, and ∫0T‍(Ys*-Ls)dKs*+=∫0T‍(Ys*-Us)dKs*-=0.
One definesτ*=inf{s∈[0,T],Ys*=Us} and θ*=inf{s∈[0,T],Ys*=Ls}.
ThenY0*=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
(3.6)Y0*=g(xT)+∫0T‍H(s,xs,Ys*,Zs*,us*,vs*)ds+K*+-KT*--∫0T‍Zs*dBs=Yτ*∧θ**+∫0τ*∧θ*‍C(s,xs,Ys*,us*,vs*)ds+Kτ*∧θ**+-Kτ*∧θ**--∫0τ*∧θ*‍Zs*dBsu*,v*.
Since K*+ and K*- increase only when Y reaches L and U, we have Kτ*∧θ**+=Kτ*∧θ**-=0. As (∫0t‍ZrdBru*,v*)t≤T is an (ℱt,Pu*,v*)-martingale, then we get
(3.7)Y0*=Eu*,v*[Yτ*∧θ**+∫0τ*∧θ*‍C(s,xs,Ys*,us*,vs*)ds+Kτ*∧θ**+-Kτ*∧θ**--∫0τ*∧θ*‍Zs*dBsu*,v*]=Eu*,v*[Yτ*∧θ**+∫0τ*∧θ*‍C(s,xs,Ys*,us*,vs*)ds].
We know thatYτ*∧θ**=Yτ**1[τ*<θ*]+Yθ**1[θ*<τ*]+Yθ**1[θ*=τ*<T]+g(xT)1[θ*=τ*=T] and Yτ**1[τ*<θ*]=Uτ*1[τ*<θ*], Yθ**1[θ*<τ*]=Lθ*1[θ*<τ*], Yθ**1[θ*=τ*<T]=Qθ*1[θ*=τ*<T]. So,
(3.8)Y0*=Eu*,v*[∫0τ*∧θ*‍Uτ*1[θ*<τ*]+Lθ*1[θ*<τ*]+Qθ*1[θ*=τ*<T]+g(xT)1[θ*=τ*=T]Eu*,v*+∫0τ*∧θ*‍C(s,xs,Ys*,us*,vs*)ds]=J(x0,u*,τ*;v*,θ*).
Next, letvt be an admissible control, and let θ∈𝒯. We desire to show that Y0*≥J(x0,u*,τ*;v,θ). We have
(3.9)Y0*=Yτ*∧θ*+∫0τ*∧θ‍H(s,xs,Ys*,Zs*,us*,vs*)ds+Kτ*∧θ*+-∫0τ*∧θ‍Zs*dBs=Uτ*1[τ*<θ]+Yθ*1[θ<τ*]+Qθ1[θ=τ*<T]+g(xT)1[θ=τ*=T]+∫0τ*∧θ‍H(s,xs,Ys*,Zs*,us*,vs*)ds+Kτ*∧θ*+-∫0τ*∧θ‍Zs*dBs.
The payoffJ(x0,u*,τ*;v,θ) can be described by the solution of following BSDE:
(3.10)Y0=Uτ*1[τ*<θ]+Lθ1[θ<τ*<T]+Qθ1[τ*=θ<T]+g(xT)1[τ*=θ=T]+∫0τ*∧θ‍C(s,xs,Ys,us*,vs)ds-∫0τ*∧θ‍ZsdBsu*,v=Uτ*1[τ*<θ]+Lθ1[θ<τ*<T]+Qθ1[τ*=θ<T]+g(xT)1[τ*=θ=T]+∫0τ*∧θ‍H(s,xs,Ys,Zs,us*,vs)ds-∫0τ*∧θ‍ZsdBs,
then
(3.11)Y0=Eu*,v[∫0τ*∧θ‍Uτ*1[τ*<θ]+Lθ1[θ<τ*<T]+Qθ1[τ*=θ<T]+g(xT)1[τ*=θ=T]YEu*,v+∫0τ*∧θ‍H(s,xs,Ys,Zs,us*,vs)ds-∫0τ*∧θ‍ZsdBs]=Eu*,v[Uτ*1[τ*<θ]+Lθ1[θ<τ*<T]+Qθ1[τ*=θ<T]+g(xT)1[τ*=θ=T]+∫0τ*∧θ‍C(s,xs,Ys,us*,vs)ds],
and J(x0;u*,τ*;v,θ)=Y0. Thanks to H(s,xs,Ys,Zs,us*,vs*)≥H(s,xs,Ys,Zs,us*,vs), Yθ*1[θ<τ*]≥Lθ1[θ<τ*<T], and Kτ*∧θ*+≥0 by the comparison theorem of BSDEs to compare (3.9) and (3.10) to get Y0*≥Y0=J(x0;u*,τ*;v,θ).
In the same way, we can show thatY0*=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 isY0*. We have proved that
(3.12)J(x0;u*,τ*;v,θ)≤Y0*=J(x0;u*,τ*;v*,θ*)≤J(x0;u,τ;v*,θ*),
for any (u,v)∈𝒰×𝒱 and τ,θ∈𝒯. Thereby,
(3.13)Y0*≤infu∈𝒰,τ∈𝒯J(x0;u,τ;v*,θ*)≤supv∈𝒱,θ∈𝒯infu∈𝒰,τ∈𝒯J(x0;u,τ;v,θ).
On the other hand,
(3.14)Y0*≥supv∈𝒱,θ∈𝒯J(x0;u*,τ*;v,θ)≥infu∈𝒰,τ∈𝒯supv∈𝒱,θ∈𝒯J(x0;u,τ;v,θ).
Now, due to the inequality
(3.15)infu∈𝒰,τ∈𝒯supv∈𝒱,θ∈𝒯J(x0;u,τ;v,θ)≥supv∈𝒱,θ∈𝒯infu∈𝒰,τ∈𝒯J(x0;u,τ;v,θ),
we have
(3.16)Y0*=infu∈𝒰,τ∈𝒯supv∈𝒱,θ∈𝒯J(x0;u,τ;v,θ)=supv∈𝒱,θ∈𝒯infu∈𝒰,τ∈𝒯J(x0;u,τ;v,θ).
The proof is now completed.


## 4. Application

In this section, we present two examples to show the applications of Section3.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 timet at an interest rate Rt>rt, where rt is the bond rate. Then, the wealth of the investor satisfies
(4.1)-dXt=b(t,Xt,Zt)dt-dCt-ZtdWt,0≤t≤T,b(t,Xt,Zt):=-[rtXt+θtZt-(Rt-rt)(Xt-Ztσt)-],
where Zt:=σtπt,θt:=σt-1(bt-rt). bt represents the instantaneous expected return rate in stock, σt which is invertible represents the instantaneous volatility of the stock, and Ct is interpreted as a cumulative consumption process. bt, rt, Rt, and σt are all deterministic bounded functions, and σt-1 is also bounded.
An American game is a contract between a brokerc1 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 (Lt)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 Lθ. 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 Uτ, and Uτ≥Lτ. The difference Uτ-Lτ 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 Qτ1[τ<T]+ξ1[τ=T]. Naturally, Uτ≥Qτ≥Lτ. Ut, Lt, and Qt 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 timet which consists of the selection of a stopping time τ∈ℱτ (or θ∈ℱθ) and a payoff Uτ (or Lθ) on exercise if τ<θ<T (or θ<τ<T) and ξ if τ=T. Set
(4.2)S~τ∧θ=ξ1{τ=θ=T}+Qτ1{τ=θ<T}+Lθ1{θ<τ<T}+Uτ1{τ<θ<T},0≤(τ∧θ)≤T,
then the price of American game contingent claim (S~τ∧θ,0≤(τ∧θ)≤T) at time t is given by
(4.3)Xt=essinfτ∈ℱτesssupθ∈ℱθXt(τ∧θ,S~τ∧θ),
where Xt(τ∧θ,S~τ∧θ) noted by Xtτ∧θ satisfies BSDE
(4.4)-dXsτ∧θ=b(s,Xsτ∧θ,Zsτ∧θ)ds-dCs-Zsτ∧θdWs,Xτ∧θτ∧θ=S~τ∧θ.
For each (ω,t), b(t,x,z) is a convex function of (x,z). It follows from [14] that we have Xtτ∧θ=esssuprt≤βt≤RtessinfCtXβ,C,τ∧θ. Here, Xβ,C,τ∧θ satisfies
(4.5)-dXsβ,C,τ∧θ=bβ(s,Xsβ,C,τ∧θ,Zsβ,C,τ∧θ)ds-dCs-Zsτ∧θdWs,Xτ∧θβ,C,τ∧θ=S~τ∧θ,bβ(s,Xt,Zt):=-βtXt-[θt+rt-βtσt]Zt,
where βt is a bounded R-valued adapted process which can be regarded as an interest rate process in finance. So,
(4.6)Xt:=essinfτ∈ℱτesssupθ∈ℱθXt(τ∧θ,S~τ∧θ)=essinfτ∈ℱt,Ctesssupθ∈ℱt,rt≤βt≤RtXtβ,C,τ∧θ=esssupθ∈ℱt,rt≤βt≤Rtessinfτ∈ℱt,CtXtβ,C,τ∧θ.
Here, Xtβ,C:=esssupθ∈ℱt,rt≤βt≤Rtessinfτ∈ℱt,CtXtβ,C,τ∧θ. Then, from [13], there exist Ztβ,C∈H2 and Ktβ,C,+Ktβ,C,-, which are increasing adapted continuous processes with K0β,C,+=0 and K0β,C,-=0, such that (Xtβ,C,Ztβ,C,Ktβ,C,+,Ktβ,C,-) satisfies the following reflected BSDE:
(4.7)-dXsβ,C=bβ(s,Xsβ,C,Zsβ,C)ds-dCs+dKsβ,C,+-dKsβ,C,--Zsβ,CdWs,XTβ,C=ξ,0≤s≤T,
with Ut≥Xtβ,C≥Lt, 0≤t≤T, and ∫0T‍(Xtβ,C-Lt)-dKtβ,C,+=0, ∫0T‍(Ut-Xtβ,C)-dKtβ,C,-=0. Then, the stopping time τ=inf{t≤s≤T;Xsβ,C=Us}, and θ=inf{t≤s≤T;Xsβ,C=Ls}.
We formulate the pricing problem of American game option to the stochastic recursive mixed zero-sum differential game problem which was studied in Section3, 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(xt)t≤T satisfy
(4.8)dxt=xtdBt,t≤1,wheretheinitialvalueisx0.
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=xt(ut+vt). Then, by the Girsanov's theorem, we can define the probability Pu,v by
(4.9)dPu,vdP=exp{∫0T‍(us+vs)dBs-12∫0T‍(us+vs)2ds}.
Under the probability Pu,v, the process Btu,v=Bt-∫0t‍(us+vs)ds is a Brownian motion.
First, we consider the following stochastic recursive zero-sum differential game:(4.10)J(x0,u,v)=Y0=Eu,v[xT+∫0T‍min{|xt|,2}+Yt(ut+vt)dt].(Yt)0≤t≤T satisfies BSDE
(4.11)-dYs=min{|xs|,2}+Ys(us+vs)ds-ZsdBsu,v,YT=xT.
Therefore,
(4.12)H(t,x,z,Y,u,v)=Z(u+v)+min{|xt|,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
(4.13)minu∈𝒰maxv∈𝒱H(t,x,Z,Y,u,v)=maxv∈𝒱minu∈𝒰H(t,x,Z,Y,u,v)=Z+min{|xt|,2}+Y,J(x0,u*,v*)=Y0=xT+∫0T‍(Zt+min{|xt|,2}+Yt)dtv-∫0T‍ZtdBt=E[x0exp(2BT)+∫0T‍exp(Bt+12t)min{|x0exp(Bt-12t)|,2}dt].
We also can get the conclusion that the optimal game value Y0*=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 (u0*,v0*)=(0,1); when x0=2, Y0=295.6, Z0=295.6, (u0*,v0*)=(0,1); andx0=3, Y0=443.4, Z0=443.4, and (u0*,v0*)=(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:(4.14)J(x0;u,τ;v,θ)=Y0u,τ;v,θ=Eu,v[∫0τ∧θ‍[min{|xt|,2}+Yt(ut+vt)]dtEu,v+(xτ+1)I[τ<θ]+(xθ-1)I[θ<τ<T]+xTI[θ=τ]∫0τ∧θ‍].
Then, (Yt)0≤t≤(τ∧θ) satisfies the following BSDE:
(4.15)Yt=(xτ+1)I[τ<θ]+(xθ-1)I[θ<τ<T]+xTI[θ=τ]+∫tτ∧θ‍[min{|xs|,2}+Ys(us+vs)]ds-∫tτ∧θ‍ZsdBsu,v.
Therefore, H(t,x,z,Y,u,v)=Z(u+v)+min{|xt|,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
(4.16)minu∈𝒰maxv∈𝒱H(t,x,Z,Y,u,v)=maxv∈𝒱minu∈𝒰H(t,x,Z,Y,u,v)=Z+min{|xt|,2}+Y,J(x0;u*,τ;v*,θ)=Y0u*,τ;v*,θ=Yτ∧θ+∫0τ∧θ‍(Zt+min{|xt|,2}+Yt)dt-∫0τ∧θ‍ZtdBt=Yτ∧θexp(12(τ∧θ)+Bτ∧θ)+∫0τ∧θ‍min{|xt|,2}exp(12(t)+Bt)dt-∫0τ∧θ‍exp(12(t)+Bt)(Zt+Yt)dBt,
where τ*=inf{t∈[0,T],Yt≥(xt+1)}, and θ*=inf{t∈[0,T],Yt≤(xt-1)}, while (u*,τ*;v*,θ*) is the saddle point. So, the optimal value is
(4.17)J(x0;u*,τ*;v*,θ*)=Y0u*,τ*;v*,θ*=Yτ*∧θ*exp(12(τ*∧θ*)+Bτ*∧θ*)+∫0τ*∧θ*‍min{|xt|,2}exp(12t+Bt)dt-∫0τ*∧θ*‍exp(12t+Bt)(Zt+Yt)dBt=E[∫0τ*∧θ*‍x0exp(2Bτ*∧θ*)+1τ*<θ*exp(Bτ*+12τ*)-1θ*<τ*exp(Bθ*+12θ*)EE+∫0τ*∧θ*‍min{|x0exp(Bt-12t)|,2}exp(12t+Bt)dt].
We also can get the conclusion that the optimal game valueY0*=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
Y
0 stands for the optimal game value, and x0 stand for the initial value of the dynamics system.



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*Source: 718714-2012-12-25.xml*