BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH TWO DISTINCT REFLECTING BARRIERS AND QUADRATIC GROWTH GENERATOR

The adaptation is related to the natural filtration of the Brownian motion (Bt)t≤T . In 1990, Pardoux and Peng introduced the notion of nonlinear backward stochastic differential equation (BSDE), namely (1.1), and gave existence and uniqueness result in their founder paper [26]. Since then the interest in BSDEs has kept growing steadily and there have been several works on that subject. The main reason is that BSDEs are encountered in many fields of mathematics such as finance [6, 7, 31], stochastic games and optimal control [4, 10–12, 14, 15], partial differential equations and homogeneization [25, 27–29]. Further, other settings of BSDEs have been introduced. In [5], El-Karoui et al. consider one-barrier reflected BSDEs, that is, the situation where the process Y of (1.1) is forced to stay above a given barrier (Lt)t≤T . In (1.1), they add a nondecreasing continuous process (Kt)t≤T which allows us to have Y ≥ L. Their motivations are linked, on the one hand, to the pricing of American options and, on the other hand, to viscosity solutions


Introduction
A solution for a BSDE whose generator is f (t,ω, y,z) and terminal value ξ is a pair of adapted processes (Y t ,Z t ) t≤T which satisfies (1.1) The adaptation is related to the natural filtration of the Brownian motion (B t ) t≤T .
In 1990, Pardoux and Peng introduced the notion of nonlinear backward stochastic differential equation (BSDE), namely (1.1), and gave existence and uniqueness result in their founder paper [26].Since then the interest in BSDEs has kept growing steadily and there have been several works on that subject.The main reason is that BSDEs are encountered in many fields of mathematics such as finance [6,7,31], stochastic games and optimal control [4, 10-12, 14, 15], partial differential equations and homogeneization [25,[27][28][29].
Further, other settings of BSDEs have been introduced.In [5], El-Karoui et al. consider one-barrier reflected BSDEs, that is, the situation where the process Y of (1.1) is forced to stay above a given barrier (L t ) t≤T .In (1.1), they add a nondecreasing continuous process (K t ) t≤T which allows us to have Y ≥ L. Their motivations are linked, on the one hand, to the pricing of American options and, on the other hand, to viscosity solutions of PDEs with an obstacle.This work has been generalized by Cvitanić and Karatzas [3] who consider the problem where the process Y is now forced to stay between two comparable given barriers L and U (L ≤ U) (see (1.2) below).In (1.1), they add a difference of nondecreasing processes K + − K − which maintain Y between L and U. Mainly, in [3], the authors show the existence and uniqueness of a solution when f is Lipschitz and the barriers are either regular (they are semimartingales) or satisfy Mokobodski's condition which, roughly speaking, means that between L and U there exists a difference of nonnegative supermartingales.
Another direction of research which has attracted many works is the weakness of conditions under which the standard equation (1.1) has a solution.Among others, we can quote [8,19,22,23].However in all of those papers, just the one-dimensional case is considered since the results are heavily based on comparison theorem which works only in that case.In [19], Kobylanski shows the existence of a solution for (1.1) when f is continuous with quadratic growth with respect to z, that is, | f (t,ω, y,z)| ≤ C(1 So in this paper we are dealing with BSDEs with two reflecting barriers L ≤ U and a quadratic growth generator f .A solution for that equation is a quadruple of adapted processes (Y ,Z,K + ,K − ) := (Y t ,Z t ,K + t ,K − t ) t≤T with values in R 1+d+1+1 which mainly satisfies the following: K + and K − are continuous nondecreasing processes, ( Under Mokobodski's condition on the barriers, this equation has been considered by Bahlali et al. in [1].They show the existence of a solution.However the trouble is that Mokobodski's condition is not easy to check in practice.Therefore the main goal of this paper is to give conditions on the barriers L and U, as general as possible (we could say minimal) which are easy to verify in practice under which the BSDE (1.2) has a solution when f is of quadratic growth in z.We mainly show that it is enough to have the barriers completely separated, that is, for any t ≤ T, L t < U t .In addition the solution we have constructed is maximal.
BSDEs with quadratic growth coefficients and just one reflecting barrier have been considered in [20].
This paper is organized as follows.In Section 2, we recall some results related to BSDEs whose generators are continuous with linear growth.We give comparison results which play a crucial role in the proof of the main theorem.Section 3 is devoted to the proof of the main result.As in the nonreflected case, we use an exponential transform in order to obtain a BDSE whose generator satisfies a structure condition.Afterwards we show that this BSDE has a solution.Finally we obtain a solution for the initial equation by using a logarithmic transform.In addition it is maximal.The main difficulty stems from the lack of integrability of the processes.It is overcome by using a Cantor diagonal procedure.In Section 4, we use the results of Section 3 in order to solve the risk-sensitive zero-sum S. Hamadène and I. Hdhiri 3 mixed stochastic differential game.Briefly the game problem which we consider is as follows.
Assume we have a system S on which two agents c 1 and c 2 intervene.The interventions of the agents have two forms, control and stopping, that is, they control S up to the time when one of them decides to stop controlling.Their actions are not free and their advantages are antagonistic, that is, there exits a payoff J, which depends on the implemented strategies, which is a reward for c 2 and a cost for c 1 .On the other hand, the expression of J integrates sensitiveness with respect to risk of the agents, either they are risk-seeking or risk-averse, by the mean of an exponential utility function.Our objective is to show existence of a fair strategy, to be precise a saddle-point, for the game and to characterize it.
We show that with this zero-sum game problem, a BSDE with two reflecting barriers of the type studied in Section 3 is associated.This BSDE gives the value function of the game and allows us to construct a saddle-point.Finally, we deal with an example for which we give a numerical result.
In Section 5, we deal with the yield of an American game or recallable option under Knightian uncertainty and exponential utility function.We characterize the lower (resp., upper) yield by means of BSDEs with quadratic growth coefficient and we study an example for which we give also a numerical result.

Preliminaries
Throughout this paper (Ω,Ᏺ,P) is a fixed probability space in which is defined a standard d-dimensional Brownian motion B = (B t ) t≤T whose natural filtration is (F 0 t := σ{B s , s ≤ t}) t≤T ; (F t ) t≤T is the completed filtration of (F 0 t ) t≤T with the P-null sets of Ᏺ, hence (F t ) t≤T satisfies the usual conditions, that is, it is right continuous and complete.On the other hand, let (i) ᏼ be the σ-algebra on [0, T] × Ω of F t -progressively measurable sets; (ii)the set of stopping times; is finite; (vi) ᏹ the set of continuous ᏼ-measurable nondecreasing processes (K t ) t≤T such that K 0 = 0 and K T < ∞, Pa.s.Now we are given four objects: (i) a function f : [0,T] × Ω × R 1+d → R such that for any (y,z) ∈ R 1+d , the process ( f (t, y,z)) t≤T is ᏼ-measurable and ( f (t,ω,0,0)) t≤T belongs to Ᏼ Let us point out that in our setting, contrary to those of some other works on the same subject (see, e.g., [1,3,13]), we require neither that Z ∈ Ᏼ 2,d nor K ± ∈ 2 .The main reason is that in many applications where those equations rise up, such as stochastic games or mathematical finance, we do not need such properties for Z and K ± .Therefore we have adopted our setting which is less narrow than the ones of the works quoted above.Now let us recall some known results for BSDEs whose generators f are continuous with respect to (y,z) and which grow sublinearly, that is, (A1) f is continuous with respect to (y,z) and there exists a constant C such that We have the following result.
Theorem 2.1 [9].Assume that (A1) holds.Let f be another function which satisfies (A1) and such that ( f (t, y,z)) t≤T is ᏼmeasurable for any (y,z) ∈ R 1+d .If we can compare f and f , then we can compare the maximal solutions of the double-barrier reflected BSDEs associated with ( f ,ξ,L,U) and ( f ,ξ,L,U), respectively.Namely we have the following result.Proposition 2.2 [1,9].Assume that P-a.s.f (t,w, y,z) ≤ f (t,w, y,z), for any t, y, and z.

In Theorem 2.1 above we already have K ±
T < ∞Pa.s., but we do not know whether T ] is finite or not.However there is some kind of local integrability for K ± and Z. Actually we have the following.Proposition 2.3 [9].There exists an increasing sequence of stopping times (γ k ) k≥0 such that (i) Pa.s., the sequence is stationary and converges to T; (ii) the sequence depends only on L,U, and the constant C quoted in (A1); (iii) for all k ≥ 0, S. Hamadène and I. Hdhiri 5

Double-barrier reflected BSDE with quadratic growth generator
From now on, we assume that U, L, and ξ are bounded, that is, there exists a constant C ≥ 0 such that P-a.s.
To do so we are going first to show the existence of a solution for the reflected BSDE associated with data obtained by an exponential transform of ( f ,ξ,L,U).More precisely let us set m 0 = essinf t,w L t (w), M 0 = esssup t,ω U t (ω), m = exp(2Cm 0 ), and M = exp(2CM 0 ) where C is the constant of (A2 (3.4) Then from (A2), the function F satisfies the following structure condition: (A3) there exists a positive constant α ≥ C such that Now to begin with we will prove that the BSDE associated with (F,exp(2Cξ),exp(2CL), exp(2CU)) has a solution.So let us set η = exp(2Cξ), L = exp(2CL), and U = exp(2CU).Then we have the following result.
Proof.Let ρ be the function from R into R such that for any y ∈ R, ρ(y where m and M are the constants previously defined.On the other hand, let k p be a smooth function from Now let F be the function defined by F(t,ω, y,z) = F(t,ω,ρ(y),z) for any (t, y,z) ∈ [0, T] × R 1+d .We are going to show that the following reflected BSDE: ) t≤T be the maximal solution of the double-barrier reflected BSDE associated with ( F p ,η,L ,U ) which exists according to Theorem 2.1.Then we have As F p ≥ F p+1 , then according to the comparison result of Proposition 2.2 we have Y p ≥ Y p+1 , K p+ ≤ K p+1+ , and K p− ≥ K p+1− .On the other hand since L t ≤ Y p t ≤ U t and through the boundedness of L ,U , there exits an upper semicontinuous process (Y t ) t≤T such that ) t≤T be the maximal solution of the reflected BSDE associated with ( F 0 ,η,L ,U ), then we have K p − ≤ K 0 − P-a.s.In addition, since F 0 is continuous and bounded, then there exists a nondecreasing stationary sequence of stopping times (γ k ) k≥0 , depending only on L ,U , and the constant C of boundedness of F 0 and converging to T, such that The rest of the proof will be divided into five steps.
S. Hamadène and I. Hdhiri 7 Step 1.For any p ≥ 0 and k ≥ 0, there exists a constant C k which depends only on k such that Actually the sequence (K p − ) p≥0 is decreasing then . Now let us prove that there exists a constant K k which may depend on k such that (3.12) For any t ≤ T, we have But ψ < 0 and then where M 1 = sup m≤x≤M |ψ (x)|.Now by taking expectations in both handsides of (3.14) we obtain (at least after using a localization argument and Fatou's lemma) Using the facts that F satisfies (A3), F ≤ F p for all p ≥ 0, and that ψ ≤ 0 yields 8 BSDE with distinct barriers and QG generator Therefore (3.17) and then Henceforth there exists a constant K k which depends on k such that Finally taking into account (3.12), we obtain the desired result.
The following step is the main one in the proof of this theorem.
Step 2. There exists a subsequence of (Z p ) p≥0 and a process Z ∈ ᏸ 2,d such that for any Indeed for any k ≥ 0, the sequence (Z p 1 [0,γk] ) p≥0 is bounded in Ᏼ 2,d then, through the Cantor diagonal procedure, there exists a subsequence of (Z p ) p≥0 which we still denote (Z p ) p≥0 such that for any k ≥ 0, the sequence (Z p 1 [0,γk] ) p≥0 converges weakly to a process (3.24) But we have also Then uniqueness of the weak limit implies that Now for any t ≤ T, let us set Since the sequence (γ k ) k≥0 is of stationary type and for any Let p < q be two positive integers, then Y p ≥ Y q .Now by applying Itô's formula and taking expectations we obtain 10 BSDE with distinct barriers and QG generator where Since K p+ (resp., K q+ ) moves only when Y p (resp., Y q ) reaches the obstacle L , then However Y q ≤ Y p implies that {Y p = L } ⊂ {Y q = L } and therefore, since ψ (0) = 0, But ψ (x) ≥ 0 for any x ≥ 0, then J 2 (p, q) ≤ 0. In the same way we can show that J 3 (p, q) ≤ 0. Finally let us deal with J 1 (p, q).First we have (3.32) The third inequality stems from (3.5).Now since ψ (Y . Now since for any sequence (u n ) n≥0 of Ᏼ 2,1 which converges weakly to u we have u 2 ≤ liminf n→∞ u n 2 , then we obtain Step 3. The process Y is continuous.
Let k be fixed and let p, q ∈ N such that p < q.For any t ≤ T we have 2 ds ds. (3.41) Now by taking the supremum, then expectations in both hand-sides finally using the Burkholder-Davis-Gundy inequality, we obtain where δ is an appropriate real constant.Now the sequence (Z p 1 [0,γk] ) p≥0 converges to Z1 [0,γk] in Ᏼ 2,d , then there exists a subsequence of (Z p 1 [0,γk] ) p≥0 which we still denote (Z p 1 [0,γk] ) p≥0 such that (3.43) The subsequence could depend on k but this fact is irrelevant since we work at fixed k.
On the other hand, there exists a constant λ > 0 such that ] → 0 as p, q → ∞.It follows that, for any k ≥ 0 the process (Y t∧γk ) t≤T is continuous.Now since (γ k ) k≥0 is of stationary type, then the process Y is continuous.
Step 4. Construction of the processes K + and K − .
Next for any k ≥ 0 we have In addition the process (K + t ) t≤T is nondecreasing and lower semicontinuous.
Step 5.The quadruple (Y t ,Z t ,K + t ,K − t ) t≤T is a solution for the double-barrier reflected BSDE associated with ( F,η,L ,U ).
Let k be fixed and let t ≤ T. Then Now according to Dini's theorem, F p (s, y,z) converges uniformly to F(s, y,z) on compact subsets, then implies that (K + t∧γk ) t≤T and (K − t∧γk ) t≤T are lower and upper semicontinuous in the same time then they are continuous.Since (γ k ) k≥0 is of stationary type, then K + and K − are continuous on the whole interval [0, T].Now P-a.s., for ω fixed there exists k 0 (ω) such that for k ≥ k 0 we have γ k (ω) = T, therefore for k great enough we have ).The proof of this fact can also be seen in [16].
Therefore the quadruple (Y t ,Z t ,K + t ,K − t ) t≤T is a solution for (3.8).Now as t≤T is also a solution for (3.6).Finally let us show that this solution is maximal.Let ( Ȳ , Z, K+ , K− ) be another solution for (3.6) and then also for (3.8).We have for any p ≥ 0, F p ≥ F then, according to Proposition 2.2 we have Y p ≥ Ȳ .By taking the limit as p → ∞, we obtain the desired result.
We are now ready to give the main result of this section.Theorem 3.2.There exists a ᏼ-measurable process (ᐅ,ᐆ, + , − ) := (ᐅ t ,ᐆ t , + t , − t ) t≤T solution for the reflected BSDE associated with ( f ,ξ,L,U), that is, which satisfies Moreover it is maximal.
Proof.Let (Y t ,Z t ,K + t ,K − t ) t≤T be the maximal solution of (3.6) then, for any t ≤ T we have These processes are well defined since Y ≥ exp(2Cm).By using Itô's formula with LnY /2C we obtain On the other hand, we have Y ≥ exp(2Cm) then ᐆ ∈ ᏸ 2,d since Z is so, and ± ∈ ᏹ since K ± belong also to ᏹ.In addition we obviously have ) be the maximal solution of the double-barrier reflected BSDE associated with ( f ,ξ,L,U) (resp., ( f ,ξ,L,U)), then Pa.s , ᐅ ≤ ᐅ .
Actually the functions f and f satisfy (A2) with C = max{C f ,C f }.So let F and F be the functions defined as in (3.4) and associated with f and f , respectively.Therefore we have t≤T ) be the maximal solution of the double-barrier reflected BSDE associated with (F,η,L ,U ) (resp., (F ,η,L ,U )), then we have Pa.s., Y ≤ Y .Henceforth from the definition we straightforwardly have Pa.s., ᐅ ≤ ᐅ .

Connection with risk-sensitive mixed zero-sum game problem
We now deal with an application of the double-barrier reflected BSDE to solve the mixed risk-sensitive zero-sum game problem.This problem consists in a system S on which two agents c 1 and c 2 intervene.The interventions of the agents have two forms, control and stopping, that is, they control S up to the time when one of them decides to stop controlling.Their actions are not free and their advantages are antagonistic, that is, there exits a payoff J, which depends on the implemented strategies, which is a reward for c 2 and a cost for c 1 .On the other hand, the expression of J integrates sensitiveness with respect to risk of the agents, either they are risk-seeking or risk-averse.Therefore the problem is to find two pairs of strategies for the agents which in a way are fair when implemented.
Let us describe precisely the setting of the problem.Let Ꮿ be the set of continuous functions w from [0, T] into R d endowed with the uniform norm and let σ be a function from [0, T] × Ꮿ into R d×d such that (C1) σ is ᏼ-measurable, that is, for any continuous ᏼ-measurable process p = (p t ) t≤T , the process (σ(t, p t )) t≤T is ᏼ-measurable; (C2) there exists a constant k such that (i) for all t ∈ [0,T], w, w ∈ Ꮿ, |σ(t, w) − σ(t, w )| ≤ k w − w t where w t = sup s≤t | ws |, t ≤ T; (ii) for any t ≤ T, |σ(t,0)| ≤ k, σ is invertible and its inverse σ −1 is bounded.These assumptions on σ imply that the stochastic functional differential equation has a unique solution (x t ) t≤T (see, e.g., [17,30]).The process x stands for the dynamic of the evolution of S when it is not controlled.
Let us now consider U (resp., V ) a compact metric space and ᐁ (resp., ᐂ) the set of ᏼ-measurable processes u = (u t ) t≤T (resp., v = (v t ) t≤T ) with values in U (resp., V ); ᐁ (resp., ᐂ) represents the set of admissible controls for c 1 (resp., c 2 ).On the other hand, hereafter ᐁ × ᐂ is called the set of admissible controls for the agents.
Next let f and h be two measurable functions from [0,T] × Ꮿ × U × V into R d and R + , respectively, such that (C3) f and h are uniformly bounded and ᏼ-measurable, that is, for any u ∈ ᐁ and v ∈ ᐂ, the processes f (t,x,u t ,v t ) t≤T and h(t,x,u t ,v t ) t≤T are ᏼ-measurable; (C4) for all t ∈ [0,T] the mapping f (t,x,•,•) (resp., h(t,x,•,•)) which with (u,v) in U × V associates f (t,x,u,v) (resp., h(t,x,u,v)) is continuous.Now let (u,v) ∈ ᐁ × ᐂ and P u,v be the probability on (Ω,Ᏺ) defined by So according to Girsanov's theorem (see, e.g., [17,30]), for any (u,v) ∈ ᐁ × ᐂ, the process B u,v := (B t − t 0 σ −1 (s,x) f (s,x,u s ,v s )ds) t≤T is a Brownian motion on (Ω,Ᏺ,P u,v ) and x is a weak solution of the following functional differential equation: When the agent c 1 (resp., c 2 ) acts on S with a control u (resp., v), its dynamic of evolution has the same law as the one of x under P u,v .In a way, this means that the interventions of the agents generate a drift.Assume now that c 1 (resp., c 2 ) decides, on the one hand, to use the control u (resp., v) and, on the other hand, to stop controlling at a stopping time τ (resp., ν).The payment between the two agents, which is a cost for c 1 and a reward for c 2 , is given by where L,U, and ξ are those of the previous section.The constant θ is related to sensitiveness of the controllers with respect to risk.In the case when they are risk-seeking (resp., risk-averse), θ > 0 (resp., < 0).Hereafter without loss of generality, we assume θ = 1.The objective of c 1 (resp., c 2 ) is to minimize (resp., maximize) the payoff J(u,τ;v,ν).Therefore the problem is to find a strategy (u * ,τ * ;v * ,ν * ) ∈ ᐁ × -× ᐂ ×such that for any (u,τ) and (v,ν) we have This strategy is called a saddle-point for the controllers.The inequalities (4.5) show that (u * ,τ * ;v * ,ν * ) is actually fair since when c 1 (resp., c 2 ) decides to act on S with (u * ,τ * ) (resp., (v * ,ν * )), then the best that c 2 (resp., c 1 ) has to do is to act with (v * ,ν * ) (resp., (u * ,τ * )), otherwise he will earn (resp., pay) less (resp., more).The expression of J(u,τ;v,ν) can be understood in the following way: (i) h(t,x,u t ,v t ) is the instantaneous cost (resp., reward) for c 1 (resp., c 2 ); (ii) L ν is the reward (resp., cost) for c 2 (resp., c 1 ) if c 2 decides first to stop controlling at ν; (iii) U τ is the cost (resp., reward) for c 1 (resp., c 2 ) if c 1 decides first to stop controlling at τ. Now let H be the Hamiltonian function associated with this game problem, that is, the function which with (t,x, p,u,v) Hereafter we suppose Isaacs condition fulfilled, that is, Under [H1], through Benes'selection theorem [2], there exists a couple of ᏼ × Ꮾ(R d )measurable functions u * (t,x, p) and v * (t,x, p) with values respectively in U and V such that Pa.s., for any (t,u, p) In addition (see, e.g., [4] for that fact) we have Let us now consider the function H * which with (t,x, p) ∈ [0,T] × Ꮿ × R d associates: The function p → H * (t,x, p) + (1/2)|p| 2 is continuous and at most with quadratic growth with respect to p since f , h, and σ −1 are bounded.The following theorem is the main result of this section.
In addition (u * ,τ * ;v * ,ν * ) is a saddle-point strategy for the risk-sensitive mixed zero-sum differential game.
Proof.According to Theorem 3.2, the double-barrier reflected BSDE associated with Then for any t ≤ T we have For t ≤ T let us set Ȳ * t = exp(Y * t ).Using Itô's formula we obtain Then where s )ds}, t ≤ T, then Y is bounded since Ȳ * and h are so.On the other hand It ô's formula implies S. Hamadène and I. Hdhiri 19 . In addition we have, The process K * ,+ (resp., K * ,− ) is increasing only when Y * = L (resp., Y * = U), then it does not increase between 0 and τ * ∧ ν * .It follows that ).On the other hand, the process ( ) t≤T is a P u * ,v * -martingale.Indeed by the Burkholder-Davis-Gundy inequality we obtain since Y * , σ −1 , and f are bounded.Then for any k ≥ 0, we have But the process Y * is bounded, then from Lebesgue's dominated convergence theorem we obtain since But the process Y u is bounded, then using the Lebesgue-dominated convergence theorem we obtain  previous proof the maximality of the solution has not been used, imply that the solution of the BSDE associated with (H * (t,x,z) + (1/2)|z| 2 ,ξ,L,U) is unique.
We are now going to deal with an example for which we give also a numerical treatment.The simulations are made on the ground of a paper by Mémin et al. [24].Basically in that work, the theoretical BSDE is associated with a discrete BSDE where the Brownian motion is replaced by a weighted sum of independent Bernoulli symmetric random variables.Then the discrete BSDE is solved backwardly.Finally with that solution, a process is constructed which converges uniformly to the theoretical solution of the BSDE.
Example 4.3.Assume that T = 1 and d = 1 and let us set L t = 0.5 ∧ |B t |, U t = max{3, |B t | ∧ 2} for any t ≤ 1 and ξ = |B 1 | ∧ 1.On the other hand, let x = (x t ) t≤T be the process such that for any t ≤ 1, , and let f (resp., h) be the function defined by ). Obviously in that case Isaacs condition is satisfied and the function H * of (4.8) is given by Let (Y t ,Z t ,K + t ,K − t ) t≤1 be the solution of the double-barrier reflected BSDE associated with (H * (t,x t ,z) + (1/2)|z| 2 ,ξ,L,U).The following drawing is a simulation of a path of Y .
Figure 4.1 shows the simulation of a path of Y .

Application in finance
We now deal with the problem of the yield of an American game, or recallable, option under Knightian uncertainty.First let us briefly describe that kind of option.
In a financial market assume we have a risky asset whose dynamic is given by dS t = S t μdt + νdB t , t ≤ T. (5.1) For the sake of simplicity we assume ν, μ > 0 are real constants and (B t ) t≤T is a 1-dimensional Brownian motion on the probability space (Ω,Ᏺ,P).
The usual American option is a contract between a trader and a broker who are, respectively, the buyer and the seller of the option.The trader pays a premium and obtains the right to ask for a wealth (L t ) t≤T when he decides within a period of time [0, T].The process L is called the payoff of the option (e.g., L t = (S t − K) + where S is the dynamic of the asset which bears the option and K the strike) and T its maturity.So the main problems are the value of the option, the optimal time when the trader should exercise his option, the existence of a hedging strategy for the broker, and so forth.In the framework of those options, the broker has no right other than to provide the wealth for the trader when the latter decides to exercise his option.
An American game option is a standard American option where the broker is allowed to cancel the contract which binds him to the trader.In a way, he recalls the option.But in that case, he pays what the trader of the option would have earned if he had exercised his option at the same time plus a money penalty.
The motivations of those options is that insurance companies, which usually are the sellers of options, try to protect themselves against some events which could threaten their existence when facing charges which they cannot predict and bear.
The problem of pricing a recallable option in a complete financial market has been considered and solved in [8].In this work we just focus on the yield of such an option under Knightian uncertainty.
So assume we have a trader c 1 who buys the recallable option and a broker c 2 who sells it.On the other hand, suppose that the option is on the risky asset whose dynamic is given by (S t ) t≤T .As said previously, if (i) the trader decides first to exercise the option at a stopping time σ, he makes a profit which is equal to e Lσ (e.g., L t = (S t − K) + where K is the strike); (ii) the broker decides first to cancel the option at τ, he pays the trader an amount equal to e Uτ ; (iii) both decide to go up to the maturity T of the option, c 1 earns e ξ paid by c 2 .The processes L, U and the random variable ξ are the same as in the previous section.On the other hand the difference U − L is the money penalty that c 2 pays for his decision to cancel the option.Therefore the yield of the option when c 1 (resp., c 2 ) exercises (resp., cancels) at σ (resp., τ) is given by (5.2) S. Hamadène and I. Hdhiri 25 However rationality of the behaviors of the broker and the trader, implies that the yield of the option is given by J(τ,σ). (5.3) Knightian uncertainty assumes that we are not sure that the probability under which the market will evolve is P. It could be P but it could also be another probability not far from P. Let P ϑ be such a probability.The parameter ϑ = (ϑ s ) s≤T is a ᏼ-measurable stochastic process valued in a compact set [−κ,κ]; κ is called the degree of Knightian uncertainty and the set of those ϑ s is denoted by Θ. Closeness of P ϑ to P leads us to assume that P ϑ is absolutely continuous with respect to P and its density function is given by ν −1 ϑ s 2 ds . (5.4) Therefore the process (S t ) t≤T of (5.1) satisfies the following: for any t ≤ T, dS t = μ − ϑ t S t dt + νS t dB ϑ t , ( where (B ϑ t = B t + t 0 ν −1 ϑ s ) t≤T is a Brownian motion under P ϑ .In a way this means that the yield of the risky asset in a short time interval dt given Ᏺ t is equal to (μ − ϑ t )S t dt.For more detail on Knightian uncertainty one can refer to [18].Now once again the behavior of the trader as well as that of the broker are rational, therefore the minimum and maximum yields of the option are given respectively by (5.7) Obviously the functions which with z associate H(t,θ 1 (z),z) and H(t,θ 2 (z),z), respectively, are continuous with linear growth.Then we have the following result whose proof is a direct consequence of Theorem 4.1.

. 20 )
Thus, it remains to show that E[K p+ γk ] < ∞.But using It ô's formula with exp(Y p t∧γk ) and taking t = T, we obtain that for all k, p ≥ 0,
2,1 ; (ii) a terminal value ξ which is a random variable F T -measurable and E[|ξ| 2 ] < ∞; (iii) two processes U := (U t ) t≤T and L := (L t ) t≤T which belong to 2 and satisfy L t < U t , for all t ≤ T, and L T ≤ ξ ≤ U T .A solution for the two reflecting barriers BSDE associated with ( f ,ξ,L,U) is a quadruple of ᏼ-measurable processes (Y ,Z,K + ,K− ):=(Y t ,Z t ,K + t ,K − t ) t≤T with values in R 1+d+1+14 BSDE with distinct barriers and QG generator such that ∀t ≤ T, |ξ| + sup Therefore it is easily seen that F p is continuous and bounded.In addition through (3.5) and the definition of k p , the sequence ( F p ) p≥0 is decreasing and lim p→∞ F p = F.