Double barrier reflected BSDEs with stochastic Lipschitz coefficient

This paper proves the existence and uniqueness of a solution to doubly reflected backward stochastic differential equations where the coefficient is stochastic Lipschitz, by means of the penalization method.


Introduction
Backward Stochastic Differential Equations (BSDEs) were introduced (in the nonlinear case) by Pardoux and Peng [21]. Precisely, given a data (ξ, f ) of a square integrable random variable ξ and a progressively measurable function f , a solution to BSDE associated with data (ξ, f ) is a pair of F t -adapted processes (Y, Z) satisfying These equations have attracted great interest due to their connections with mathematical finance [9,10], stochastic control and stochastic games [3,17] and partial differential equations [20,22].
In their seminal paper [21], Pardoux and Peng generalized such equations to the Lipschitz condition and proved existence and uniqueness results in a Brownian framework. Moreover, many efforts have been made to relax the Lipschitz condition on the coefficient. In this context, Bender and Kohlmann [2] considered the so-called stochastic Lipschitz condition introduced by El Karoui and Huang [8].
Further, El Karoui et al. [11] have introduced the notion of reflected BSDEs (RB-SDEs in short), which is a BSDE but the solution is forced to stay above a lower barrier. In detail, a solution to such equations is a triple of processes (Y, Z, K) satisfying (2) where L, the so-called barrier, is a given stochastic process. The role of the continuous increasing process K is to push the state process upward with the minimal energy, in order to keep it above L; in this sense, it satisfies T 0 (Y t − L t )dK t = 0. The authors have proved that equation (2) has a unique solution under square integrability of the terminal condition ξ and the barrier L, and the Lipschitz property of the coefficient f .
RBSDEs have been proven to be powerful tools in mathematical finance [10], mixed game problems [6], providing a probabilistic formula for the viscosity solution to an obstacle problem for a class of parabolic partial differential equations [11].
Later, Cvitanic and Karatzas [6] studied doubly reflected BSDEs (DRBSDEs in short). A solution to such an equation related to a generator f , a terminal condition ξ and two barriers L and U is a quadruple of (Y, Z, K + , K − ) which satisfies In this case, a solution Y has to remain between the lower barrier L and upper barrier U . This is achieved by the cumulative action of two continuous, increasing reflecting processes K ± . The authors proved the existence and uniqueness of the solution when f (t, ω, y, z) is Lipschitz on (y, z) uniformly in (t, ω). At the same time, one of the barriers L or U is regular or they satisfy the so-called Mokobodski condition, which turns out into the existence of a difference of a non-negative supermartingales between L and U . In addition, many efforts have been made to relax the conditions on f , L and U [1,15,16,18,19,27,29] or to deal with other issues [5,[12][13][14]24]. Let us have a look at the pricing problem of an American game option driven by Black-Scholes market model which is given by the following system of stochastic differential equations where r(t) is the interest rate process, θ(t) is the risk premium process, σ(t) is the volatility process of the market. The fair price of the American game option is defined by where ℑ [0,T ] is the collection of all stopping times τ with values between 0 and T , and J is a Payoff given by Here r(t), σ(t) and θ(t) are stochastic, moreover they are not bounded in general. So the existence results of Cvitanic and Karatzas [6], Li and Shi [19] with completely separated barriers cannot be applied.
Motivated by the above works, the purpose of the present paper is to consider a class of DRBSDEs driven by a Brownian motion with stochastic Lipschitz coefficient. We try to get the existence and uniqueness of solutions to those DRBSDEs by means of the penalization method and the fixed point theorem. Furthermore, the comparison theorem for the solutions to DRBSDEs will be established.
The paper is organized as follows: in Section 2, we give some notations and assumptions needed in this paper. In Section 3, we establish the a priori estimates of solutions to DRBSDEs. In Section 4, we prove the existence and uniqueness of solutions to DRBSDEs via penalization method when one barrier is regular, in the first subsection, then we study the case when the barriers are completely separated, in the second subsection. In Section 5, we give the comparison theorem for the solutions to DRBSDEs. Finally, an Appendix is devoted to the special case of RBSDEs with lower barrier when the generator only depends on y; furthermore, the corresponding comparison theorem will be established under the stochastic Lipschitz coefficient.

Notations
Let (Ω, F , (F t ) t≤T , P) be a filtered probability space. Let (B t ) t≤T be a d-dimensional Brownian motion. We assume that (F t ) t≤T is the standard filtration generated by the Brownian motion (B t ) t≤T .
We will denote by |.| the Euclidian norm on R d . Let's introduce some spaces: • L 2 is the space of R-valued and F T -measurable random variables ξ such that • S 2 is the space of R-valued and F t -progressively measurable processes (K t ) t≤T such that Let β > 0 and (a t ) t≤T be a non-negative F t -adapted process. We define the increasing continuous process A(t) = t 0 a 2 (s)ds, for all t ≤ T , and introduce the following spaces: • L 2 (β, a) is the space of R-valued and F T -measurable random variables ξ such that ξ 2 β = E e βA(T ) |ξ| 2 < +∞.
• S 2 (β, a) is the space of R-valued and F t -adapted continuous processes • S 2,a (β, a) is the space of R-valued and F t -adapted processes (Y t ) t≤T such that • H 2 (β, a) is the space of R d -valued and F t -progressively measurable processes (Z t ) t≤T such that We consider the following conditions: (H1) The terminal condition ξ ∈ L 2 (β, a).
The two reflecting barriers L and U are two F t -adapted and continuous real-valued processes which satisfy where L + and U − are the positive and negative parts of L and U , respectively.
(H6) U is regular: i.e., there exists a sequence of (U n ) n≥0 such that where the processes u n and v n are F t -adapted such that sup n≥0 sup 0≤t≤T u n (t) Definition 1. Let β > 0 and a be a non-negative F t -adapted process. A solution to DRBSDE is a quadruple (Y, Z, K + , K − ) satisfying (3) such that • K ± ∈ S 2 are two continuous and increasing processes with K ± 0 = 0.
3 A priori estimate Lemma 1. Let β > 0 be large enough and assume (H1)−(H6) hold. Let (Y, Z, K + , K − ) ∈ (S 2 (β, a) ∩ S 2,a (β, a)) × H 2 (β, a) × S 2 × S 2 be a solution to DRBSDE with data (ξ, f, L, U ). Then there exists a constant C β depending only on β such that Proof. Applying Itô's formula and Young's inequality, combined with the stochastic Lipschitz assumption (H2) we can write Using the fact that dK + Taking expectation on both sides above, we get and by the Burkholder-Davis-Gundy's inequality we obtain To conclude, we now give an estimate of K + T 2 and K − T 2 . From the equation and the stochastic Lipschitz property (H2), we have Combining this with (7), we derive that The desired result is obtained by estimates (6), (8) and (9).

The obstacle U is regular
In this part, we apply the penalization method and the fixed point theorem to give the existence of the solution to the DRBSDE (3). We first consider the special case when the generator does not depend on (y, z): Theorem 1. Assume that g a ∈ H 2 (β, a) and (H1)-(H6) hold. Then, the doubly reflected BSDE (3) with data (ξ, g, L, U ) has a unique solution (Y, Z, For all n ∈ N, let (Y n , Z n , K n+ ) be the F t -adapted process with values in (S 2 (β, a) ∩ S 2,a (β, a)) × H 2 (β, a) × S 2 being a solution to the reflected BSDE with data (ξ, We denote K n− t := n t 0 (Y n s − U s ) + ds and g n (s, y) := g(s) − n(y − U s ) + . We have divided the proof of Theorem 1 into sequence of lemmas.
Proof. For all n, m ≥ 0, let (Y n,m , Z n,m ) be the solution to the following BSDE For n ≥ 0, let D n be the class of F t -progressively measurable process taking values in [0, n]. For ν ∈ D n and λ ∈ D m we denote R t = e − t 0 (ν(s)+λ(s))ds . Applying Itô's formula to R tȲ n,m t and using the same arguments as on page 2042 of [6], one can show thatȲ From the assumption (H6)(ii), we can writeȲ n,m There exists a positive constant C ′ β depending only on β such that for all Proof. Itô's formula implies for t ≤ T : Here we used the fact that −nY n s (Y n s − U s ) + ≤ nU − (Y n s − U s ) + and dK n+ s = 1 {Y n s =Ls} dK n+ s . We conclude, by the Burkholder-Davis-Gundy's inequality, that In the same way as (9), we can prove that We obtain the desired result.
for all t ≤ T . Therefore, there exist processes Y and K + such that, as n → +∞, for all t ≤ T , Y n t ց Y t and K n+ t ր K + t . Since the process K + is continuous, it follows by Dini's theorem that Let ( Y n , Z n , K n ) be the solution to the following Reflected BSDE associated with (ξ, g − n(y − U ), L): The comparison Theorem 5 shows that Y n ≤ Y n and d K n ≤ dK n+ ≤ dK + . Let τ ≤ T be a stopping time. Then we can write In addition, Consequently, Therefore, Y τ ≤ U τ P-a.s. We deduce, from Theorem 86 page 220 in Dellacherie and Meyer [7], that Y t ≤ U t for all t ≤ T P-a.s and then e βA(t) (Y n t − U t ) + ց 0 for all t ≤ T P-a.s. By Dini's theorem, we have sup 0≤t≤T e βA(t) (Y n t − U t ) + ց 0 P-a.s. and the result follows from the Lebesgue's dominated convergence theorem. Lemma 6. There exist two processes (Z t ) t≤T and (K − t ) t≤T such that Moreover, Proof. For all n ≥ p ≥ 0 and t ≤ T , applying Itô's formula and taking expectation yields that Therefore, using Lemmas 2 and 5, we obtain It follows that (Z n ) n≥0 is a Cauchy sequence in complete space H 2 (β, a). Then there exists an F t -progressively measurable process (Z t ) t≤T such that the sequence (Z n ) n≥0 tends toward Z in H 2 (β, a). On the other hand, by the Burkholder-Davis-Gundy's inequality, one can derive that where c is a universal non-negative constant. It follows that

Now, we set
One can show, at least for a subsequence (which we still index by n), that The proof is completed.
Proof of Theorem 1. Obviously, the process (Y t , Z t , K + t , K − t ) t≤T satisfies, for all t ≤ T , Since Y n t ≥ L t and from Lemma 5 we have L t ≤ Y t ≤ U t . In the following, we want to show that Let ω ∈ Ω be fixed. It follows from Lemma 4 that, for any ε > 0, there exists n(ω) On the other hand, since the function (Y t (ω) − L t (ω)) t≤T is continuous, then there exists a sequence of non-negative step functions (f m (ω)) m≥0 which converges uni- and, since (f m (ω)) m≥0 is a step function, Therefore, we have The arbitrariness of ε and Y ≥ L, show that T 0 (Y t − L t )dK + t = 0. Further, by Lemma 4 and the result treated on p. 465 of Saisho [25] we can write ≤ 0 for each n ≥ 0 P-a.s. and for each n, m ≥ 0, n = m, Then we have Combining (13) and (14), we get is the solution to (3) associated to the data (ξ, g, L, U ).
We can now state the main result: Proof. Given (φ, ψ) ∈ B 2 , consider the following DRBSDE : From (H2) and (H3), we have It follows from (H4) that f a ∈ H 2 (β, a) and then (15) has a unique solution (Y, Z, We define a mapping ϕ : Applying Itô's formula to e βA(t) |∆Y t | 2 and taking expectation we have We have used the fact that ∆Y s d(∆K + s − ∆K − s ) ≤ 0. Choosing αβ = 4 and β > 5, β . It follows that ϕ is a strict contraction mapping on B 2 and then ϕ has a unique fixed point which is the solution to the DRBSDE (3).
Remark 1. If we consider U = +∞, we obtain the BSDE with one continuous reflecting barrier L, then we proved the existence and uniqueness of the solution to RBSDE (2) by means of a penalization method. Before this work, Wen Lü [26] showed the existence and uniqueness result for this class of equations via the Snell envelope notion.

Completely separated barriers
In this section we will prove the existence of solution to (3) when the barriers are completely separated, i.e., L t < U t , ∀t ≤ T . Then (H7) there exists a continuous semimartingale with h ∈ H 2 (0, a) and V ± ∈ S 2 (V ± 0 = 0) are two nondecreasing continuous processes, such that We will show the existence by the general penalization method. We first consider the special case when the generator does not depend on (y, z): Let (Y n , Z n ) ∈ (S 2 (β, a) ∩ S 2,a (β, a)) × H 2 (β, a) be solution to the following BSDE We denote K n+ t := n t 0 (Y n s − L s ) − ds, K n− t := n t 0 (Y n s − U s ) + ds, K n t = K n+ t − K n− t and f n (s, y) = f (s) − n(y − U s ) + + n(y − L s ) − . Now let us derive the uniform a priori estimates of (Y n , Z n , K n+ , K n− ).

Lemma 7.
There exists a positive constant κ independent of n such that, ∀n ≥ 0, Proof. Consider the RBSDE with data (ξ, f, L). That is, From Appendix A there exists a unique triplet of processes (Y , Z, K) ∈ (S 2 (β, a) ∩ S 2,a (β, a)) × H 2 (β, a) × S 2 being the solution to RBSDE (18). We consider the penalization equation associated with the RBSDE (18), for n ∈ N, Similarly, we consider the RBSDE with data (ξ, f, U ). There exists a unique triplet of processes (Y , Z, K) ∈ (S 2 (β, a) ∩ S 2,a (β, a)) × H 2 (β, a) × S 2 , which satisfies By the penalization equation associated with the RBSDE (19) Z n s dB s and the Remark 2, we deduce that Y n t ≥ Y t for all t ≤ T . Then we can write On the other hand, using Itô's formula and taking expectation implies for t ≤ T : Now we need to estimate E[ For this, let us consider the following stopping times   Since Y , L and U are continuous processes and L < U , τ l < τ l+1 on the set {τ l+1 < T }. In addition the sequence (τ l ) l≥0 is of stationary type (i.e. ∀ω ∈ Ω, there exists l 0 (ω) such that τ l0 (ω) = T ). Indeed, let us set G = {ω ∈ Ω, τ l (ω) < T, l ≥ 0}, and we will show that P(G) = 0. We assume that P(G) > 0, therefore for ω ∈ G, we have Y τ 2l+1 ≤ L τ 2l+1 and Y τ 2l ≥ U τ 2l . Since (τ l ) l≥0 is nondecreasing sequence then τ l ր τ , hence U τ ≤ Y τ ≤ L τ which is contradiction since L < U . We deduce that P(G) = 0. Obviously Y n ≥ L on the interval [τ 2l , τ 2l+1 ], then the BSDE (17) becomes On the other hand, using the assumption (H7), we get From (22) and the definition of process H we obtain By summing in l, using the fact that Y n ≤ U on the interval [τ 2l+1 , τ 2l+2 ], we can In the same way, we obtain Combining (23), (24) with (21), we obtain the desired result.
Proof. Consider the following BSDE for each n ∈ N By the Remark 2, we have Y n t ≥ Y n t for all t ≤ T . Let ν be a stopping time such that ν ≤ T . Then By the fact that L is uniformly continuous on [0, T ], it can be shown that the sequence (X n t ) n≥1 uniformly converges in t, and the same for (X n− t ) n≥1 . Lebesgue's dominated convergence theorem implies that From the fact that Y n t ≥ Y n t for all t ≤ T we deduce that Similarly to proof of the Lemma 5, we can obtain Proof. Itô's formula implies that Hence

Lemma 8 implies that
From the equation we can conclude that The proof is completed.
Proof. From Lemma 9, we obtain that there exists an adapted process (Y, Z, K) ∈ (S 2 (β, a) ∩ S 2,a (β, a)) × H 2 (β, a) × S 2 such that Then, passing to the limit as n → +∞ in the equation Let τ ≤ T be a stopping time, by Lemma 7 we obtain that the sequences K n± τ are bounded in L 2 , consequently, there exist F τ -measurable random variables K ± τ in L 2 , such that there exist the subsequences of K n± τ weakly converging in K ± τ . Now we set K τ = K + τ − K − τ . By [28] (Mazu's Lemma, p. 120), there exists, for every n ∈ N, an integer N ≥ n and a convex combination N j=n ζ τ,n j (K ± τ ) j with ζ τ,n j ≥ 0 and Denoting K n τ = K n+ τ − K n− τ , it follows that Thanks to (30), we have K n τ − K τ L 2 < ε for all ε > 0. Therefore Combining (32) and (33), we obtain K τ = K τ a.s. Therefore, from Theorem 86, p. 220 in [7] we have K t = K t for all t ≤ T . On the other hand, (31) implies that, for τ = T , there exists a subsequence of K n+ T tȲ + s (ζ sȲs + η sZs + δ s )ds − 2