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

Mathematical Problems in Engineering Volume 2012, Article ID 718714,15pages doi:10.1155/2012/718714

Research Article

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

Lifeng Wei

and Zhen Wu

1School of Mathematical Sciences, Ocean University of China, Qingdao 266003, China

2School of Mathematics, Shandong University, Jinan 250100, China

Correspondence should be addressed to Zhen Wu,[email protected] Received 4 October 2012; Accepted 10 December 2012

Academic Editor: Guangchen Wang

Copyrightq2012 L. Wei and Z. Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 equationsBSDEsand 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 shorthad been intro- duced by Pardoux and Peng1, who proved the existence and uniqueness of adapted solu- tions under suitable assumptions. Independently, Duffie and Epstein2introduced BSDE from economic background. In2, they presented a stochastic differential recursive utility which is an extension of the standard additive utility with the instantaneous utility depend- ing 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 variableZ. From mathematical point of view, the result in1is more general. Then, El Karoui et al.3and Cvitanic and Karatzas4generalized, respectively, the results to BSDEs with reflection at one barrier and two barriersupper and lower.

BSDE plays an important role in the theory of stochastic differential game. Under the notable Isaacs’s condition, Hamad`ene and Lepeltier 5 obtained the existence result of a saddle point for zero-sum stochastic differential game with payoff

Ju, v E^u,v T

t

fs, xs, us, vsds gxT

. 1.1

Using a maximum principle approach, Wang and Yu6,7 proved the existence and uni- queness of an equilibrium point. We note that the cost function in5is not recursive, and the game system in6,7is a BSDE. In8, 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 followingSection 2, we proved the existence of a saddle point for the stochastic recursive zero-sum differential game problem and also got the optimal payofffunction by the solution of one specific BSDE. Here, the generator of the BSDE contains the main variable solutionyt, and we extend the result in5to the recursive case which has much more significance in economics theory.

Then, inSection 3we 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 Hamad`ene9. 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 in9.

In fact, the recursive mixed zero-sum game problem has wide application back- ground 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 ourSection 3. To show the application of this kind of problem and our motivation to study our recursivemixedgame 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, inSection 4, we also give another example of the recursive mixed zero-sum game problem, for which the explicit saddle point and optimal payofffunction 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 spaceΩ,F, P. LetFt_t≥0be the completed natural filtration ofB_t. Moreover,

iCis the space of continuous functions from0, TtoR^m; iiP is theσ-algebra on0, T×ΩofFt-progressively sets;

iiifor any stopping time ν,Tν is the set ofFt-measurable stopping timeτ such that P-a.s. ν≤τ ≤T;T0will simply be denoted by T;

ivH^2,k is the set of P-measurable processes ω ωt_t≤T, R^k-valued, and square integrable with respect todt⊗dP;

vS² is the set of P-measurable and continuous processesω ωt_t≤T, such that Esup_t≤T|ω_t|²<∞.

Them×mmatrixσ σijsatisfies the following:

ifor any 1≤i,j ≤m, σ_ij is progressively measurable;

iifor anyt, x∈0, T× C, the matrixσt, xis invertible;

iiithere exists a constantsK such that|σt, x−σt, x| ≤ K|x−x|t and|σt, x| ≤ K1 |x|t.

Then, the equation

xtx0

_t

0

σs, xsdBs, t≤T 2.1

has a unique solutionxt.

Now, we consider a compact metric spaceAresp.B, andUresp.V is the space ofP-measurable processes u : ut_t≤T resp.v : vt_t≤T with values inAresp.B. Let Φ:0, T× C × U × V → R^mbe such that

ifor anyt, x∈0, T× C, the mappingu, v → Φt, x, u, vis continuous;

iifor anyu, v∈A×B, the function Φ·, x·, u, visP-measurable;

iiithere exists a constantK such that|Φt, x, u, v| ≤K1 |x|tfor anyt,x,u, andv;

ivthere exists a constantMsuch that|σ⁻¹t, xΦt, x, u, v| ≤Mfor anyt,x,u, andv.

Foru, v∈ U × V, we define the measureP^u,vas dP^u,v

dP exp _T

0

σ⁻¹s, xsΦs, xs, u_s, v_sdBs−1 2

_T

0

σ⁻¹s, xsΦs, xs, u_s, v_s²ds

. 2.2

Thanks to Girsanov’s theorem, under the probabilityP^u,v, the process

B_t^u,vBt− _t

0

σ⁻¹s, xsΦs, xs, us, vsds, t≤T, 2.3

is a Brownian motion, and for this stochastic differential equation

x_tx_{0 t}

0

Φs, x_s, u_s, v_sds _t

0

σs, xsdB^u,vs , t≤T, 2.4

xt_t≤Tis a weak solution.

Suppose that we have a system whose evolution is described by the processxt_t≤T. On that system, two agentsc₁andc₂intervene. A control action forc₁resp.c₂is a process

u ut_t≤T resp.v vt_t≤Tbelonging toUresp.V. TherebyUresp.Vis called the set of admissible controls forc₁resp.c₂. Whenc₁andc₂act with, respectively,uandv, the law of the dynamics of the system is the same as the one ofxunderP^u,v. The two agents have no influence on the system, and they act to protect their advantages by means ofu∈ Uand v∈ Vvia the probabilityP^u,v.

In order to define the payoff, we introduce two functions Ct, x, y, u, v and gx satisfying the following assumption: there existsL > 0, for allx, x ∈ H^2,mand Y, Y ∈ S², such that

Ct, xt, Y_t, u, v−C

t, x_t, Y_t, u, v ≤Lx_t−x_t, Yt−Y_t Ct, xt, Yt, u, v−C

t, xt, Y_t, u, v

≤L

Yt−Y_t ²,

2.5

andgxis measurable, Lipschitz continuous function with respect tox. The payoffJx0, u, v is given byJx0, u, v Y0, whereY satisfies the following BSDE:

−dYsCs, xs, Y_s, u_s, v_sds−Z_sdB^u,v_s ,

YTgxT. 2.6

From the result in10, there exists a unique solutionY, Zforu, v. The agentc₁ wishes to minimize this payoff, and the agentc2 wishes to maximize the same payoff. We investigate the existence of a saddle point for the game, more precisely a pairu^∗, v^∗of strategies, such thatJx0, u^∗, v≤Jx0, u^∗, v^∗≤Jx0, u, v^∗for eachu, v∈ U × V.

Fort, x, Y, Z, u, v∈0, T× C ×R×R^m× U × V, we introduce the Hamiltonian by Ht, x, Y, Z, u, v Zσ⁻¹t, xΦt, x, u, v Ct, x, Y, u, v, 2.7

and we say that the Isaacs’ condition holds if fort, x, Y, Z∈0, T× C ×R×R^m, maxv∈V min

u∈U Ht, x, Y, Z, u, v min

u∈U max

v∈V Ht, x, Y, Z, u, v. 2.8 We suppose now that the Isaacs’ condition is satisfied. By a selection theoremsee Benes11, there existsu^∗: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 Ht, x, Y, Z, u^∗t, x, Y, Z, v^∗t, x, Y, Zis Lipschitz inZand monotone inYlike the functionC.

Now we give the main result of this section.

Theorem 2.1. Y^∗, Z^∗is the solution of the following BSDE:

−dY_s^∗Hs, xs, Y_s^∗, Z^∗_s, u^∗s, x, Y^∗, Z^∗, v^∗s, x, Y^∗, Z^∗ds−Z^∗_sdB_s,

Y_T^∗gxT. 2.10

Then,Y₀^∗is the optimal payoffJx0, u^∗, v^∗, and the pairu^∗, v^∗is the saddle point for this recursive game.

Proof. We consider the following BSDE:

Y_t^∗gxT _T

t

Hs, xs, Y_s^∗, Z^∗_s, u^∗t, x, Y^∗, Z^∗, v^∗t, x, Y^∗, Z^∗ds− _T

t

Z^∗_sdBs. 2.11

Thanks to Theorem 2.1 in 10, the equation has a unique solutionY^∗, Z^∗. Because Y₀^∗ is deterministic, so

Y₀^∗E^u∗,v∗ Y₀^∗ E^u∗,v∗

gxT

_T

0

Hs, xs, Y_s^∗, Z_s^∗, u^∗t, x, Y^∗, Z^∗, v^∗t, x, Y^∗, Z^∗ds− _T

0

Z^∗_sdBs

E^u∗,v∗

gxT _T

0

Cs, xs, Y_s^∗, u^∗_s, v^∗_sds− _T

0

Z^∗_sdB^u_s^∗,v∗

.

2.12

We can getY₀^∗Jx0, u^∗, v^∗.

For anyu∈ U, v∈ V, then we let

YtgxT _T

t

Cs, xs, Ys, u^∗_s, vsds− _T

t

ZsdB^u_s^∗,v gxT

_T

t

Hs, xs, Y_s, Z_s, u^∗_s, v_sds− _T

t

Z_sdB_s,

Y_t gxT _T

t

C

s, xs, Y_s, us, v_s^∗ ds− _T

t

Z_sdB_s^u,v∗ gxT

_T

t

H

s, x_s, Y_s, Z_s, u_s, v^∗_s ds− _T

t

Z_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 Jx0, u^∗, v ≤ Jx0, u^∗, v^∗≤Jx0, u, v^∗ Y₀andu^∗, 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 byxt_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 ofu∈ UforC1andv∈ VforC2via the probabilityP^u,vin 2.2. The coupleu, v∈ U × Vis called an admissible control for the game. Both controllers

also have the possibility to stop controlling atτ forC1andθforC2;τ andθare elements of Twhich is the class of allFt-stopping time. In such a case, the game stops. The controlling action is not free, and it corresponds to the actions ofC₁andC₂. A payoffis described by the following BSDE:

Y_t^u,τ;v,θ U_τ1_τ<θ L_θ1θ<τ<T Q_τ1_τθ<T gxT1_{τθT τ∧θ}

t

C

s, xs, Y_s^u,τ;v,θ, us, vs

ds−

_τ∧θ

t

ZsdB_s^u,v,

3.1

and the payoffis given by Jx0;u, τ;v, θ Y₀^u,τ;v,θ

E^u,v _τ∧θ

0

C

s, xs, Y_s^u,τ;v,θ, us, vs

ds Uτ1_τ<θ Lθ1θ<τ<T

Qτ1_τθ<T gxT1τθT

,

3.2

where theUt_t≤T,Lt_t≤T, andQt_t≤T are processes ofS²such thatLt≤Qt≤Ut. The action ofC1 is to minimize the payoff, and the action ofC2 is to maximize the payoff. Their terms can be understood as

iCs, x, Y, u, vis the instantaneous reward forC₁ and cost forC₂; iiU_τ is the cost forC₁ and forC₂ if C₁ decides to stop first the game;

iiiL_θis the reward forC₂and cost forC₁ ifC₂ decides stop first the game.

The problem is to find a saddle point strategyone should say a fair strategyfor the controllers, that is, a strategyu^∗, τ^∗;v^∗, θ^∗such that

Jx0;u^∗, τ^∗;v, θ≤Jx0;u^∗, τ^∗;v^∗, θ^∗≤Jx0;u, τ;v^∗, θ^∗, 3.3

for anyu, τ;v, θ∈ U × T × V × T.

Like inSection 2, we also define the Hamiltonian associated with this mixed stochastic game problem by Ht, x, Y, Z, u, v, and thanks to the Benes’s solution 11, there exist u^∗t, x, Y, Zandv^∗t, x, Y, Zsatisfying

Ht, x, Y, Z, u^∗t, x, Y, Z, v^∗t, x, Y, Z max

v∈V min

u∈U

Zσ⁻¹t, xΦt, x, u, v Ct, x, Y, u, v min

u∈U max

v∈V

Zσ⁻¹t, xΦt, x, u, v Ct, x, Y, u, v . 3.4 It is easy to know thatHt, x, Y, Z, u, vis Lipschitz inZand monotone inY.

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^∗gxT _T

t

Hs, xs, Y_s^∗, Z_s^∗, u^∗_s, v^∗_sds

K^∗ _T −K_t^∗ −

K_T^∗−−K_t^∗− − _T

t

Z^∗_sdB_s, 3.5

satisfying for all 0≤t≤T, L_t≤Y_t^∗≤U_t, and_T

0Y_s^∗−L_sdK^∗ _{s T}

0Y_s^∗−U_sdK_s^∗−0.

One definesτ^∗inf{s∈0, T, Y_s^∗Us}andθ^∗inf{s∈0, T, Y_s^∗Ls}.

ThenY₀^∗Jx0;u^∗, τ^∗;v^∗, θ^∗,u^∗, τ^∗;v^∗, θ^∗is the saddle point strategy.

Proof. It is easy to know that the reflected BSDE3.5has a unique solutionY^∗, Z^∗, K^∗ , K^∗−, then we have

Y₀^∗gxT _T

0

Hs, xs, Y_s^∗, Z^∗_s, u^∗_s, v^∗_sds K^∗ −K^∗−_T − _T

0

Z_s^∗dB_s Y_τ^∗∗∧θ^∗

_τ∗∧θ^∗ 0

Cs, xs, Y_s^∗, u^∗_s, v^∗_sds K_τ^∗ ∗∧θ^∗−K^∗−_τ∗∧θ^∗− _τ∗∧θ^∗

0

Z_s^∗dB_s^u∗,v∗.

3.6

SinceK^∗ andK^∗− increase only whenY reachesLandU, we haveK_τ^∗ _∗∧θ∗ K_τ^∗−_∗∧θ∗ 0. As _t

0ZrdB_r^u∗,v∗_t≤Tis anFt, P^u∗,v∗-martingale, then we get

Y₀^∗E^u∗,v∗

Y_τ^∗∗∧θ^∗

_τ∗∧θ^∗ 0

Cs, xs, Y_s^∗, u^∗_s, v^∗_sds K_τ^∗ ∗∧θ^∗−K^∗−_τ∗∧θ^∗− _τ∗∧θ^∗

0

Z_s^∗dB_s^u∗,v∗

E^u∗,v∗

Y_τ^∗∗∧θ^∗

_τ∗∧θ^∗ 0

Cs, xs, 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 gxT1_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^∗_<TQ_θ^∗1_θ^∗_τ^∗_<T. So,

Y₀^∗E^u∗,v∗

Uτ^∗1_θ^∗<τ^∗ Lθ^∗1_θ^∗<τ^∗ Qθ^∗1_θ^∗_τ^∗_<T gxT1θ^∗τ^∗T

_τ^∗_∧θ^∗

0

Cs, xs, Y_s^∗, u^∗_s, v^∗_sds

Jx0, u^∗, τ^∗;v^∗, θ^∗.

3.8

Next, let vt be an admissible control, and let θ ∈ T. We desire to show that Y₀^∗ ≥ Jx0, u^∗, τ^∗;v, θ. We have

Y₀^∗Y_τ^∗∗∧θ

_τ^∗_∧θ

0

Hs, xs, Y_s^∗, Z_s^∗, u^∗_s, v^∗_sds K_τ^∗ ∗∧θ− _τ^∗_∧θ

0

Z_s^∗dBs

U_τ∗1_τ∗<θ Y_θ^∗1_θ<τ∗ Q_θ1_θτ∗<T gxT1_θτ^∗_{T τ}^∗_∧θ

0

Hs, xs, Y_s^∗, Z^∗_s, u^∗_s, v^∗_sds K_τ^∗ ∗∧θ− _τ^∗_∧θ

0

Z^∗_sdBs.

3.9

The payoffJx0, u^∗, τ^∗;v, θcan be described by the solution of following BSDE:

Y₀U_τ^∗1_τ^∗_<θ L_θ1_θ<τ^∗_<T Q_θ1_τ^∗_θ<T gxT1τ^∗θT

_τ^∗_∧θ

0

Cs, xs, Ys, u^∗_s, vsds− _τ^∗_∧θ

0

ZsdBs^u∗,v

U_τ^∗1_τ^∗_<θ L_θ1_θ<τ^∗_<T Q_θ1_τ^∗_θ<T gxT1τ^∗θT

_τ^∗_∧θ

0

Hs, xs, Ys, Zs, u^∗_s, vsds− _τ^∗_∧θ

0

ZsdBs,

3.10

then

Y0E^u∗,v

Uτ^∗1_τ^∗_<θ Lθ1_θ<τ^∗_<T Qθ1_τ^∗_θ<T gxT1τ^∗θT

_τ^∗_∧θ

0

Hs, xs, Ys, Zs, u^∗_s, vsds− _τ^∗_∧θ

0

ZsdBs

E^u∗,v

U_τ^∗1_τ^∗_<θ L_θ1_θ<τ^∗_<T Q_θ1_τ^∗_θ<T gxT1τ^∗θT

_τ^∗_∧θ

0

Cs, xs, Y_s, u^∗_s, v_sds

, 3.11 and Jx0;u^∗, τ^∗;v, θ Y0. Thanks to Hs, xs, Ys, Zs, u^∗_s, v^∗_s ≥ Hs, xs, Ys, Zs, u^∗_s, vs, Y_θ^∗1_θ<τ^∗ ≥Lθ1_θ<τ^∗_<T, andK_τ^∗ ∗∧θ≥ 0 by the comparison theorem of BSDEs to compare3.9 and3.10to getY₀^∗≥Y₀Jx0;u^∗, τ^∗;v, θ.

In the same way, we can show thatY₀^∗Jx0;u^∗, τ^∗;v^∗, θ^∗≤Jx0;u, τ;v^∗, θ^∗for any τ ∈ Tand 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

Jx0;u^∗, τ^∗;v, θ≤Y₀^∗Jx0;u^∗, τ^∗;v^∗, θ^∗≤Jx0;u, τ;v^∗, θ^∗, 3.12

for anyu, v∈ U × Vandτ, θ∈ T. Thereby, Y₀^∗≤ inf

u∈U,τ∈TJx0;u, τ;v^∗, θ^∗≤ sup

v∈V,θ∈T inf

u∈U,τ∈TJx0;u, τ;v, θ. 3.13

On the other hand,

Y₀^∗≥ sup

v∈V,θ∈TJx0;u^∗, τ^∗;v, θ≥ inf

u∈U,τ∈T sup

v∈V,θ∈TJx0;u, τ;v, θ. 3.14

Now, due to the inequality

u∈U,τ∈Tinf sup

v∈V,θ∈TJx0;u, τ;v, θ≥ sup

v∈V,θ∈T inf

u∈U,τ∈TJx0;u, τ;v, θ, 3.15

we have

Y₀^∗ inf

u∈U,τ∈T sup

v∈V,θ∈TJx0;u, τ;v, θ sup

v∈V,θ∈T inf

u∈U,τ∈TJx0;u, τ;v, θ. 3.16

The proof is now completed.

4. Application

In this section, we present two examples to show the applications ofSection 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 Hamad`ene9proved 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 timetat an interest rate Rt> rt, wherertis the bond rate. Then, the wealth of the investor satisfies

−dXtbt, Xt, Ztdt−dCt−ZtdWt, 0≤t≤T, bt, Xt, Z_t:−

r_tX_t θ_tZ_t−Rt−r_t

X_t−Z_t σ_t

₋ ,

4.1

where Zt : σtπt, θt : σ_t⁻¹bt−rt. bt 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. bt,rt,Rt, andσt are all deterministic bounded functions, andσ_t⁻¹is 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 amountthe price of the optionwhich guarantees a payment ofLt_t≤T. The trader can exercise whenever he decides before the maturityT of the option. Thus, if the trader decides to exercise atθ, he gets the amountLθ. On the other hand, the broker is allowed to cancel the contract. Therefore, if he choosesτ as the contract cancellation time, he pays the amountUτ, andUτ ≥Lτ. The differenceU_τ−L_τis the premium that the broker pays for his decision to cancel the contract.

Ifc1 andc2decide together to stop the contract at the timeτ, thenc2gets a reward equal to Qτ1_τ<T ξ1_τT. Naturally,Uτ ≥Qτ ≥Lτ.Ut,Lt, andQtare 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 timeτ ∈ Fτ orθ∈ Fθand a payoffUτ orLθ on exercise ifτ < θ < Torθ < τ < TandξifτT. Set

S_τ∧θξ1_{τθT} Q_τ1_{τθ<T} L_θ1{θ<τ<T} U_τ1{τ<θ<T}, 0≤τ∧θ≤T, 4.2

then the price of American game contingent claimS_τ∧θ, 0≤τ∧θ≤Tat timetis given by

X_tess inf

τ∈Fτ

ess sup

θ∈Fθ

X_t

τ∧θ,S_τ∧θ

, 4.3

whereX_tτ∧θ,S_τ∧θnoted byX_t^τ∧θsatisfies BSDE

−dX^τ∧θ_s b

s, X_s^τ∧θ, Z^τ∧θ_s ds−dCs−Z_s^τ∧θdWs, X_τ∧θ^τ∧θS_τ∧θ.

4.4

For eachω, t,bt, x, zis a convex function ofx, z. It follows from14that we haveX_t^τ∧θ ess sup_r

t≤βt≤Rtess inf_CtX^β,C,τ∧θ. Here,X^β,C,τ∧θsatisfies

−dX^β,C,τ∧θs b^β

s, X_s^β,C,τ∧θ, Z_s^β,C,τ∧θ

ds−dCs−Z_s^τ∧θdWs, X_τ∧θ^β,C,τ∧θS_τ∧θ,

b^βs, Xt, Z_t:−βtX_t−

θ_t r_t−β_t σ_t

Z_t,

4.5

whereβtis a boundedR-valued adapted process which can be regarded as an interest rate process in finance. So,

Xt:ess inf

τ∈Fτ

ess sup

θ∈Fθ

Xt

τ∧θ,S_τ∧θ ess inf

τ∈Ft,Ct

ess sup

θ∈Ft,rt≤βt≤Rt

X^β,C,τ∧θ_t ess sup

θ∈Ft,rt≤βt≤Rt

ess inf

τ∈Ft,Ct

X_t^β,C,τ∧θ.

4.6

Here,X^β,C_t : ess sup_θ∈F

t,rt≤βt≤Rtess inf_τ∈Ft,CtX_t^β,C,τ∧θ. Then, from13, there existZ_t^β,C ∈ H² and K^β,C, _t K^β,C,−_t , which are increasing adapted continuous processes with K₀^β,C, 0 and K₀^β,C,−0, such thatX^β,C_t , Z_t^β,C, K_t^β,C, , K_t^β,C,−satisfies the following reflected BSDE:

−dXs^β,Cb^β

s, X_s^β,C, Z_s^β,C

ds−dC_s dK^β,C, _s −dK^β,C,−_s −Z^β,C_s dW_s, X^β,C_T ξ, 0≤s≤T,

4.7

withUt≥X^β,C_t ≥Lt, 0≤t≤T, and_T

0X_t^β,C−Lt⁻dK^β,C, _t 0,_T

0Ut−X^β,C_t ⁻dK_t^β,C,−0. Then, the stopping timeτ inf{t≤s≤T; X_s^β,CUs}, andθinf{t≤s≤T; X_s^β,CLs}.

We formulate the pricing problem of American game option to the stochastic recursive mixed zero-sum differential game problem which was studied inSection 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 systemxt_t≤Tsatisfy

dxtxtdBt, t≤1, where the initial value is x0. 4.8

The control action forc1resp.c2isuresp.vwhich belongs toUresp.V. TheUis0,1, and theVis0,1, while the functionΦ xtut vt. Then, by the Girsanov’s theorem, we can define the probabilityP^u,vby

dP^u,v dP exp

_T

0

us v_sdBs−1 2

_T

0

us v_s²ds

. 4.9

Under the probabilityP^u,v, the processB_t^u,vB_t−_t

0us v_sdsis a Brownian motion.

First, we consider the following stochastic recursive zero-sum differential game:

Jx0, u, v Y0E^u,v

xT

_T

0

min{|xt|,2} Ytut vtdt

. 4.10

Yt_0≤t≤Tsatisfies BSDE

−dYsmin{|xs|,2} Y_sus v_sds−Z_sdB^u,v_s ,

Y_T x_T. 4.11

Therefore,

Ht, x, z, Y, u, v Zu v min{|xt|,2} Yu v, 4.12

and obviously, the Isaacs condition is satisfied withu^∗1_{Z Y≤0}, v^∗1_{Z Y≥0}. It follows that minu∈U max

v∈V Ht, x, Z, Y, u, v max

v∈V min

u∈U Ht, x, Z, Y, u, v Z min{|xt|,2} Y, Jx0, u^∗, v^∗ Y0

x_{T T}

0

Zt min{|xt|,2} Y_tdtv− _T

0

Z_tdB_t E

x0exp2BT _T

0

exp

Bt 1 2t

min

x0exp

Bt−1 2t

,2

dt

.

4.13

We also can get the conclusion that the optimal game valueY₀^∗Jx0, u^∗, v^∗is an increasing function with the initial value of the dynamics systemx₀from the previous representation.

Now, we give the numerical simulation and drawFigure 1 to show this point. LetT 2, whenx₀ 1, the optimal game valueY₀ 147.8,Z₀ 147.8 and the saddle point strategy u^∗₀, v^∗₀ 0,1; whenx₀2,Y₀295.6,Z₀ 295.6,u^∗₀, v^∗₀ 0,1; andx₀ 3,Y₀ 443.4, Z0 443.4, andu^∗₀, v^∗₀ 0,1.Y0 is increasing function ofx0 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,v _τ∧θ

0

min{|xt|,2} Ytut vtdt

xτ 1Iτ<θ xθ−1Iθ<τ<T xTI_θτ

.

4.14

1 2 3 4 5 6 7 100

200 300 400 500 600 700 800 900 1000 1100

y(0)

x(0) Y: 147.8X: 1

Y: 295.6X: 2

Y: 443.4X: 3

Figure 1:Y0stands for the optimal game value, andx0stand for the initial value of the dynamics system.

Then,Yt_{0≤t≤τ∧θ}satisfies the following BSDE:

Yt xτ 1Iτ<θ xθ−1Iθ<τ<T xTI_{θτ τ∧θ}

t

min{|xs|,2} Y_sus v_sds− _τ∧θ

t

Z_sdB^u,v_s .

4.15

Therefore, Ht, x, z, Y, u, v Zu v min{|xt|,2} Yu v, and obviously, the Isaacs condition is satisfied withu^∗1_{Z Y≤0}, v^∗1_{Z Y≥0}. It follows that

minu∈U max

v∈V Ht, x, Z, Y, u, v max

v∈V min

u∈U Ht, x, Z, Y, u, v Z min{|xt|,2} Y, Jx0;u^∗, τ;v^∗, θ Y₀^{u∗,τ;v∗,θ}

Y_{τ∧θ τ∧θ}

0

Zt min{|xt|,2} Ytdt− _τ∧θ

0

ZtdBt

Y_τ∧θexp 1

2τ∧θ B_τ∧θ

_τ∧θ

0

min{|xt|,2}exp 1

2t Bt

dt

− _τ∧θ

0

exp 1

2t B_t

Zt Y_tdBt,

4.16