Reflected BSDEs with two optional barriers and monotone coefficient on general filtered space

We consider reflected backward stochastic differential equations with two optional barriers of class (D) satisfying Mokobodzki's separation condition and coefficient which is only continuous and non-increasing. We assume that data are merely integrable and the terminal time is an arbitrary (possibly infinite) stopping time. We study the problem of existence and uniqueness of solutions, and their connections with the value process in nonlinear Dynkin games.


Introduction
Let F = (F t ) t≥0 be a general filtration satisfying merely the usual conditions and T be an arbitrary (possibly infinite) F-stopping time. We also assume as given an F T -measurable random variable ξ, a real function f defined on Ω × R + × R, which is F-progressively measurable with respect to first two variables, and two F-optional processes L, U satisfying some separation condition. In the present paper, we consider reflected backward stochastic differential equations (RBSDE for short) which informally can be written in the form (1. 1) In (1.1), O (resp. M loc , V p ) is the space of F-optional processes (resp. local Fmartingales, finite variation F-predictable processes). We study the problem of existence and uniqueness of solutions to (1.1) and connections of (1.1) with nonlinear Dynkin games.
The notion of BSDEs with two reflecting barriers was introduced in 1996 by Cvitanic and Karatzas [7]. They considered bounded terminal time T , continuous barriers L, U and filtration F generated by a Brownian motion. In that case (1.1) may be formulated rigorously as follows: where O c (resp. V c ) is the subspace of O (resp. V p ) consisting of continuous processes. In [7] it is assumed that the generator f is Lipschitz continuous with respect to y, z and the data are in L 2 , that is ξ, f (·, 0, 0), sup t≤T |L t | and sup t≤T |U t | are square-integrable. Note that in the special case of Brownian filtration each local F-martingale M is of the form M = Z dB, which allows one to consider f depending also on Z. In [7] an existence and uniqueness result for (1.2) is proved. Moreover, it is shown there that the solution is linked with Dynkin game via the formula holding for any stopping time α ∈ [0, T ]. A similar result was obtained in 1997 by El Karoui et al. [12] in the case of one barrier.
The theory of RBSDEs has been significantly developed over the last two decades and assumptions from the paper by Cvitanic and Karatzas [7] were successively weakened. We will provide a brief review of the literature to show the main directions of relaxing of the standard assumptions considered in the pioneering paper by Cvitanic and Karatzas. The case of càdlàg barriers is considered in [17,22,29]. RBSDEs with monotone generator satisfying weak growth condition are studied in [22,28,34] and we refer the reader to [5,21,22,34] for equations with L p -data for p ∈ [1,2]. Equations with Brownian-Poisson filtration and càdlàg barriers were studied in [16,18,19,20,36]. The case of general, right-continuous filtration F, monotone generator f and L 1 -data was studied in [23]. Equations with T being an arbitrary stopping time were studied in [35], in [1] (Brownian-Poisson filtration) and [24] (general filtration).
In most of the existing papers on RBSDEs càdlàg barriers are considered, and there are only few papers dealing with non-càdlàg case. Such equations with L 2 -data and Lipschitz continuous generator were studied in [30] (Brownian filtration), in [13,14,31] (Brownian-Poisson filtration) and [3,4,15] (general filtration). RBSDEs with L 1 -data and optional barriers were considered only in [25,26] in case of Brownian filtration and bounded terminal time.
As already mentioned, in the present paper we study the existence and uniqueness of solutions of class (D) to RBSDEs (1.1) with general filtration F and possibly infinite F-stopping time T . We assume that L and U are F-optional processes of class (D) satisfying Mokobodzki's condition and such that lim sup (1.4) As for ξ, f and V we will assume that they satisfies the following conditions.
In Section 3, for f satisfying (H1)-(H4) we introduce the notion of nonlinear fexpectation E f α,β : L 1 (Ω, F β , P ) → L 1 (Ω, F α , P ), associated with BSDE (1.1) with no reflection, and we prove its basic properties. Here α ≤ β are stopping times. In Section 4, we give a definition of a solution to (1.1). By the definition, we are looking for a triple (Y, M, R) ∈ O × M loc × V p such that Y is a regulated (làdlàg) process such that for every a ≥ 0, and moreover, Y satisfies the terminal condition of the form We also require that R is minimal in the sense that for every a ≥ 0, and R * ,+ (resp. R * ,− ) is the càdlàg part of the process R + (resp. R − ). We prove that there exists at most one solution (Y, M, R) to RBSDE (1.1) such that Y is of class (D). In Section 5, we prove that under (H1)-(H4) there exists a solution (Y, M, R) to RBSDE (1.1) with one reflecting lower barrier L such that Y is of class (D). In this case R is an increasing process. We also show that for every stopping time α ≤ T , Let us stress here that in general ER T and E T 0 |f (r, Y r )| dr are infinite. We give necessary and sufficient condition for which these integrals are finite. We show that if this is the case, then In Section 6, we are focused on our main goal, i.e. equation (1.1). We first show that each solution to (1.1) such that Y is of class (D) admits the representation where ρ, δ are the so called stopping systems (see Section 6), and Using this representation we prove a stability result for (1.1). To prove the existence of a solution, we consider the nonlinear decoupling system introduced in the linear case (f ≡ 0) by Bismut [6]. Since, as we mentioned before, the integral in (1.5) may be infinite, we reformulate (1.5) as a the following system of RBSDEs with one reflecting lower barrier: (1.6) Using the results of Section 5 we prove that under (H1)-(H4), (1.4) and Mokobodzki's condition (the existence of a special semimartingale between the barriers) there exists a solution ( is a solution to (1.1). We also give a necessary and sufficient condition under which E|R| T and E T 0 |f (r, Y r )| dr are finite. Finally, using our stability result, we show that there exists a solution (Y, M, R) to (1.1) such that Y is of class (D) even if f does not satisfy (H1), i.e. under (H2)-(H4), (1.4) and Mokobodzki's condition. It is worth pointing here that in general, without (H1) imposed on f there is no solution to equation of type (1.1) with no reflection. In other words, we show that for the existence of solutions to reflected BSDEs weaker assumptions on f are needed than for the existence of solutions to related BSDEs.

Notation and standing assumptions
Let a ≥ 0. We say that a function y : [0, a] → R is regulated if the limit y t+ = lim u↓t y u exists for every t ∈ [0, a), and the limit y s− = lim u↑s y u exists for every s ∈ (0, a]. For any regulated function y on [0, a] we set ∆ + y t = y t+ − y t if 0 ≤ t < a, and ∆ − y s = y s − y s− if 0 < s ≤ a. It is known that each regulated function is bounded and has at most countably many discontinuities (see, e.g., [10, Chapter 2, Corollary 2.2]). For x ∈ R, we set sgn(x) = 1 x =0 x/|x|. Let (Ω, F, P ) be a probability space, F = (F t ) t≥0 be a filtration satisfying the usual conditions and let T be an F-stopping time. For a ≥ 0, we set T a = T ∧ a. For fixed stopping times σ, τ we denote by T σ,τ the set of all F-stopping times taking values in [σ, τ ]. We alsoset T = T 0,T and T τ = T τ,T . An increasing sequence {τ k } ⊂ T is called a chain (on [0, T ]) if ∀ω ∈ Ω ∃n ∈ N ∀k ≥ n τ k (ω) = T.
We say that an F-progressively measurable process X is of class (D) if the family {X τ , τ ∈ T , τ < ∞} is uniformly integrable. We equip the space of processes of class (D) with the norm X 1 = sup τ ∈T , τ <∞ E|X τ |. In the sequel in case if X ∞ is not defined we set X ∞ = 0.
We denote by M (resp. M loc ) the set of all F-martingales (resp. local martingales) such that M 0 = 0, and by V (resp. V + ) the space of all F-progressively measurable processes of finite variation (resp. increasing) such that V 0 = 0. V 1 is the set of processes V ∈ V (resp. V ∈ V + ) such that E|V | T < ∞, where |V | T stands for the total variation of V on [0, T ]. V p (resp. V + p ) is the space of all predictable V ∈ V (resp. V ∈ V + ). For V ∈ V, by V * we denote the càdlàg part of the process V , and by V d its purely jumping part consisting of right jumps, i.e.
In the whole paper all relations between random variables are understood to hold P -a.s. For processes X and Y , For a process X, we set − → X s = lim sup r↑s X r , ← − X s = lim sup r↓s X r , X − → s = lim inf r↑s X r and X ← − s = lim inf r↓s X r , s ∈ [0, T a ], a ≥ 0. By [8,Theorem 90,page 143], if X is an optional process of class (D), then ← − X , X ← − are progressively measurable, and − → X , X − → are predictable processes.
In the whole paper, L and U are F-adapted optional processes of class (D), and ξ is F T -measurable random variable. We always assume that (1.4) is satisfied. The generator (coefficient) is a map which is F-progressively measurable for fixed y. As for V , we always assume that V ∈ V.
3 BSDEs and nonlinear f -expectation Ta 0 |f (r, Y r )| dr < +∞ for every a ≥ 0, (c) for every a ≥ 0, Remark 3.2. Existence, uniqueness and some other properties of solutions to equation BSDE T (ξ, f + dV ) we will use later on follow from [24] (see also [27]). In these papers it is assumed that V is càdlàg but the results of [24,27] may be applied to the case when V is regulated by a simple change of variables. Indeed, if (Ȳ ,M ) is a solution to Proof. By the Meyer-Tanaka formula, which when combined with the previous inequality and standard localization argument gives the desired result.
We now introduce the notion of nonlinear expectation.
In Proposition 3.5 we gather some properties of nonlinear f -expectation which will be needed later on. These properties are direct consequences of the existence, uniqueness, stability, and comparison theorems for BSDEs proved in [24]. Properties (i)-(iii) below were proved in [27,Proposition 5.6].
(i) Let ξ ∈ L 1 (Ω, F β , P ) and V be an F-adapted finite variation process such that V α = 0 and E|V | α,β < ∞. Then there exists a unique solution (X, We will also need the following lemma. By (3.1), (H2) and Lemma 3.3, 2) and the Lebesgue dominated convergence theorem, By what has been proved, Since ε > 0 was arbitrary, this proves the lemma.

Definition of a solution and a comparison result
Proof. By (H2) and the fact that By the minimality condition for R 1 , R 2 and the assumption that L 1 ≤ L 2 and U 1 ≤ U 2 , and By [25,Corollary A.5], for all a ≥ 0 and stopping times σ, τ ∈ T Ta such that σ ≤ τ we have By the above inequality, (4.1)-(4.3) and the assumption that dV 1 ≤ dV 2 , we have Taking the expectation and then letting k → ∞ we obtain Therefore, by the Section Theorem (see, e.g., [8,Chapter IV,Theorem 86

Existence of a solution with one reflecting barrier
We will need the following additional assumption.
(H5) There exists a process X such that L ≤ X, E T 0 f − (r, X r ) dr < ∞ and X is a difference of two supermartingales of class (D) on [0, T ].

Snell envelope of an optional process of class (D)
In this section, we recall some properties of Snell envelope of an optional process of class (D). In the whole section, we assume that L is an optional process of class (D) defined on [0, T ]. Given α ∈ T , we set By [9, page 417], Y is the smallest supermartingale majorizing L.
Proof. By the considerations preceding Lemma 5.1 we may and will assume that Y is positive. By the fact that Y is a supermartingale and L ≤ Y , Since Y is positive, for every τ ∈ T α,σ we have Hence ess sup which when combined with (5.1) proves the lemma.

Existence results for linear equations
Proof. For a stopping time α ∈ T , we put Observe that is the Snell envelope of the processL t : page 417], S is a supermartingale of class (D). By the Mertens decomposition theorem (see [32]), there exist K ∈ V + p and M ∈ M loc such that Y is a regulated process of class (D). Moreover, by (5.3), Y ≥ L, and from (5.4) it follows that for every a ≥ 0, We will show that (Y, M, K) is a solution of RBSDE T (ξ, f + dV, L). To check this it remains to show the minimality condition (b) and condition (d) of Definition 4.1 are satisfied. Applying Lemma 5.1 and [15, On the other hand, for τ = T ,

A priori estimates and Snell envelope representation for solutions to nonlinear RBSDEs
For an F-adapted regulated process Y , we set Assume also that ξ ∈ L 1 and V ∈ V 1 . Then for every α ∈ T , where X t = X 0 + C t + H t , t ≥ 0, is the Doob-Meyer decomposition of the process X appearing in (H5). Moreover, M is a uniformly integrable martingale and The existence of the solution follows from [24, Proposition 2.7]. By Proposition 4.3, Y ≥ Y . By (H5), there exists a process X such that X is of class (D), X ≥ L and E T 0 f − (r, X r ) dr < ∞. There exist processes H ∈ M loc and C ∈ V 1 p such that for every a ≥ 0, This equation can be rewritten as By (H5), (5.9) and [24, Theorem 2.8], for some C 1 > 0. We will show that EK T < ∞. For every a ≥ 0 we have Let {τ k } be a localizing sequence on [0, T a ] for the local martingale M . By (5.11) with T a replaced by τ k , Taking the expectation and letting k → ∞ and then a → ∞, we obtain By (H2), By the above inequality and (5.10), so M is a uniformly integrable martingale. This completes the proof.

Existence results for nonlinear RBSDEs
Proposition 5.5. Assume that (H1), (H2), (H4) are satisfied and there is a measurable λ : [0, ∞) → R + such that ∞ 0 λ(r) dr < ∞ and Then there exists a unique solution (Y, The existence of the solution follows from [24, Proposition 2.7]. Next, for each n ≥ 1, we define (Y n , M n , K n ) to be a solution of RBSDE T (ξ, f n + dV, L) with such that Y n is of class (D Since Y n is of class (D), it follows from (H1) and the assumption that f is Lipschitz continuous that Consider the monotone subsequences {Y 2k } k∈N and {Y 2k+1 } k∈N of {Y n } n∈N , and setȲ = lim k→∞ Y 2k , Y = lim k→∞ Y 2k+1 . We shall show thatȲ is a solution of RBSDE T (ξ,f n + dV, L) withf (r) = f (r, Y r ) and Y is a solution of RBSDE T (ξ, f + dV, L) with f (r) = f (r,Ȳ r ). By (5.14),Ȳ and Y are of class (D) and lim a→∞ȲTa = lim a→∞ Y Ta = ξ. By (H2) and (5.14), for each n ≥ 1, By similar arguments we have, for every σ ∈ T T , By [15, Lemma 3.1, Lemma 3.2], the processesȲ , Y are regulated and there exist K, K ∈ V + p andM , M ∈ M loc such that for all a ≥ 0 and t ∈ [0, T a ], Moreover, by [ for all a ≥ 0 and t ∈ [0, T a ]. Therefore (Ȳ ,M ,K) is a solution of RBSDE T (ξ,f +dV, L), and (Y , M , K) is a solution of RBSDE T (ξ, f + dV, L). We will show thatȲ = Y . By [25,Corollary A.5], for all a ≥ 0 and stopping times σ, τ ∈ T Ta such that σ ≤ τ we have By the minimality condition forK and the fact that L ≤Ȳ , and By the assumption on f we have that Let {τ k } be a localizing sequence on [0, T a ] for the local martingale · σ sgn {Ȳ r− >Y r− } d(M r − M r ). By (5.23) with τ replaced by τ k ≥ σ, for k ∈ N. Taking the expectation and then letting k → ∞ and using Fubini's theorem we get for all a ≥ 0. Applying now Gronwall's lemma with σ replaced by σ ∧ t and t ∈ [0, T a ] we get Proof. For each n ≥ 1 let where 0 ≤ c n ≤ 1, c n (r) ր 1 as n → ∞ and T 0 c n (r) dr < ∞. Let (Y n , M n , K n ) be a solution of RBSDE T (ξ, f n + dV, L) such that Y n is of class (D). It is easy to check that for each n ≥ 1 the hypotheses (H1), (H2) and (H4) are satisfied and for all t ∈ [0, T ] and y 1 , y 2 ∈ R, |f n (t, y 1 ) − f (t, y 2 )| ≤ c n n |y 1 − y 2 |.
Therefore the existence of such solutions follows from Proposition 5.5. Moreover, since for each n ≥ 1, f n ≤ f n+1 , we have that Y n ≤ Y n+1 by Proposition 4.3. Set Y = sup n≥1 Y n . Then Y is of class (D) and lim a→∞ Y Ta = ξ. To see this, consider the solution (X, H, C) of RBSDE T (ξ, 0, L) such that X is of class (D). The existence of the solution follows from Proposition 5.5. We know that X is supermartingale majorizing L and since f (t, y) ≥ g(t) for t ∈ [0, T ] and y ∈ R, we have E T 0 f − (r, X r ) dr < ∞. Moreover, EC T < ∞. By the definition of a solution of RBSDE we know that for every a ≥ 0,

This equation can be rewritten in the form
Let (X,H) be a solution of BSDE T (ξ,f + dV + + dC ′ ,+ ) such thatX is of class (D). The existence of the solution follows from [24,Proposition 2.7]. Note that (X, H) is a solution of BSDE T (ξ, f + dV + dC ′ ). By Proposition 4.3,X ≥ X, soX ≥ L. Moreover, the triple (X,H, 0) is a solution of RBSDE T (ξ,f + dV + + dC ′ ,+ ,X). Therefore, by Proposition 4.3,X ≥ Y n , n ≥ 1. We have Y 1 ≤ Y n ≤X, (5.26) so Y is of class (D) and lim a→∞ Y Ta = ξ. By (H3), f n (r, Y n r ) → f (r, Y r ) as n → ∞. Since f n ≤ f n+1 , from (H2), (5.26) and the assumption on f it also follows that Observe that {τ k } is a chain on [0, T ] and the triple (Y n , M n , K n ) is a solution of RBSDE(Y n τ k ,f n + dV ,L) on [0, τ k ]. Hence, by Proposition 5.3, for every σ ∈ T τ k , By the definition of τ k and (5.27), as n → ∞. By (5.26), (5.28), (5.29) and [25,Lemma 3.19], for σ ∈ T τ k . By [15, Lemma 3.1, Lemma 3.2], Y is regulated and there exist K k ∈ V + p and M k ∈ M loc such that for all a ≥ 0 and t ∈ τ k ∧ a], Also, by [ . Therefore, since {τ k } is a chain, we can define processes K and M on each interval [0, T a ] by putting Proof. We consider a strictly positive function g : [0, ∞) → R such that ∞ 0 g(r) dr < ∞. For each n ≥ 1, let f n (r, y) = f (r, y) ∨ (−n · g(r)), From theorem Theorem 5.6 we know that for each n ≥ 1 there exists a solution (Y n , M n , K n ) to of RBSDE T (ξ, f n +dV, L) such that Y n is of class (D). Since f n ≥ f n+1 , Y n ≥ Y n+1 by Proposition 4.3. Set Y = inf n≥1 Y n . Then Y is of class (D) and lim a→∞ Y Ta = ξ. To see this, consider a solution (X, H) to BSDE T (ξ,f + dV ) such that X is of class (D). It exists by [24,Proposition 2.7]. By Proposition 4.3, so Y is of class (D) and lim a→∞ Y Ta = ξ. By (H3), f n (r, Y n r ) → f (r, Y r ) as n → ∞. Moreover, since f n ≥ f n+1 , it follows from (H2) and (5.30) that Then {τ k } is a chain [0, T ] and (Y n , M n , K n ) is a solution to RBSDE(Y n τ k ,f n + dV ,L) on [0, τ k ]. Hence, by Proposition 5.3, for evry σ ∈ T τ k , for σ ∈ T τ k . By [15, Lemma 3.1, Lemma 3.2], Y is regulated and there exist K k ∈ V + p and M k ∈ M loc such that for all a ≥ 0 and t ∈ [0, τ k ∧ a], Moreover, by [   appearing in hypothesis (H1) one can consider the following more general condition: there exists a process S which is a difference of two supermartingales of class (D) such that However, without loss of generality one can assume that (5.34) is satisfied. Indeed, let (Y, M, K) be a solution of RBSDE T (ξ, f + dV, L). Since S is a difference of supermartingales, there exist processes H ∈ M loc and C ∈ V 1 p such that for every a ≥ 0,

Optimal stopping problem with nonlinear f -expectation
Repeating step by step the proofs of the results of Section 8 in [15] and [15, Theorem 6.1], with using Proposition 3.5 and Proposition 4.3, we get the following result.
(ii) Y is the smallest E f -supermartingale majorizing L ξ , (iii) If L is u.s.c. from the right, then Y coincides with the first component of the solution to RBSDE T (ξ, f, L).
Then, by Proposition 4.3,Ȳ ≥ Y . On the other hand, since Y ′ is an E f -supermartingale, Y ′ α = ess sup τ ≥α E f α,τ (Y ′ τ ). Therefore, by Lemma 5.9(iii) and Proposition 4.3,Ȳ = Y ′ (since Y ′ is u.s.c. from the right as an E f -supermartingale). Thus Y ′ ≤ Y , which implies that Y is the smallest E f -supermartingale majorizing L ξ . This when combined with Lemma 5.9(ii) gives the desired result.

Existence results for RBSDEs with two barriers and Dynkin games
In this section, we prove existence results for reflected BSDEs with two optional barriers satisfying Mokobodzki's condition: (H6) There exists a special semimartingale X such that L ≤ X ≤ U .
As in the case of one barrier, we first prove the existence of integrable solutions to RBSDEs with two barriers satisfying the following stronger condition: (H6*) There exists a process X being a difference of two supermartingales of class (D) on [0, T ] such that L ≤ X ≤ U and E T 0 |f (r, X r )| dr < ∞. However, we start with showing that each solution to RBSDE is the value function in a nonlinear Dynkin game. This representation will also be needed in the proof of the main result. We denote by S the set of all stopping systems and for fixed stopping times σ, γ ∈ T we denote by S σ,γ the set of stopping systems ρ = (τ, H) such that σ ≤ τ ≤ γ. We put S σ := S σ,T . Note that any stopping time τ ∈ T can be identified with a stopping system (τ, Ω). Therefore we may write T ⊂ S. For a stopping system ρ = (τ, H) and for an optional process X, we set Remark 6.3. In general, (6.1) is not true if we replace stopping systems by stopping times. The proof of (6.1) is much more simpler then the proof of the corresponding result for one barrier (Theorem 5.10). This is due to the fact that in (6.1) we can always indicate ε-optimal stopping systems regardless on the regularity of barriers L, U . These ε-optimal stopping systems ρ ε = (τ ε , H ε ), δ ε = (σ ε , G ε ) are given by the following formulas (see [14, (4.19)]), Note also that formulas of type (6.1) for linear RBSDEs (however without using the notion of RBSDEs) were proved in [2].
As for Y 2 , by Proposition 5.3, for all n ≥ 1 and σ ∈ T T , Letting n → ∞ and using (6.4) we get By [15, Lemma 3.1, Lemma 3.2], Y 1 is regulated and there exist K k ∈ V + p and M k ∈ M loc such that for every a ≥ 0, , a ≥ 0. From this and the fact that lim a→∞ Y 1 Ta = ξ it follows that (Y 1 , Z 1 , K 1 ) is a solution of RBSDE T (ξ,f +dV, L+Y 2 ). By [15, Lemma 3.1, Lemma 3.2] again, Y 2 is regulated and there exist K 2 ∈ V + p , M 2 ∈ M loc such that for every a ≥ 0, By [ We shall show that (Y, Z, R) is a solution of RBSDE T (ξ,f + dV ,L,U ). Observe that for every a ≥ 0, and lim a→∞ Y Ta = ξ. Clearly L ≤ Y ≤ U . Furthermore, R satisfies the minimality condition because for every a ≥ 0 we have is a solution to RBSDE T (ξ, f + dV, L) (resp. RBSDE T (ξ, f + dV, U )). Therefore (6.3) follows from (H2) and Proposition 5.4. Proposition 6.6. Assume that f, f n , n ≥ 1, satisfy (H1)-(H4) and f n ր f as n → ∞. Let {L n } be a sequence of optional processes of class (D) on [0, T ] such that L n ր L.
By (H2), (6.10) and the fact that f n ≤ f n+1 and f n ≤ f we have Observe that {σ k } is a chain on [0, T ]. By the definition of {σ k } and (6.15), as n → ∞. Let {γ k } be a localizing sequence on [0, T a ] for the local martingale . By (6.14) with τ replaced by Since Y and Y n are of class (D) and we know that (6.16) holds true, taking the expectation in (6.17), letting n → ∞ and then k → ∞, we obtain E|Y σ −Ỹ σ | ≤ E|Y Ta −Ỹ Ta | for a ≥ 0. Letting now a → ∞ we get E|Y σ −Ỹ σ | = 0. Therefore, by the Section Theorem (see, e.g., [8, Chapter IV, Proof. Let X be the process appearing in condition (H6). Since X is a special semimartingale, there exists an increasing sequence {γ k } ⊂ T such that X is a difference of supermartingales of class (D) on [0, γ k ] for every k ≥ 1. Let ̺ be a strcitly positive Borel measurable function on R + such that ∞ 0 ̺(t) dt < ∞, and let f n,m (t, y) = n̺(t) 1 + n̺(t) max{min{f (t, y), n}, −m}.
Observe that f n,m is increasing with respect to n and decreasing with respect to m. Moreover, f n,m (t, y) ր f m (t, y) = max{f (t, y), −m} as n → ∞ and f m (t, y) ց f (t, y) as m → ∞. LetL,Û be regulated processes defined bŷ Observe thatL ≤ L n ≤ L n+1 ≤ L ≤ U ≤ U n+1 ≤ U n ≤Û , n ≥ 1.
Since all the processes |L|, | ← − L |, |U |, |U ← − |, |Y | are of class (D) on [0, T ], using Lemma 3.6 we conclude that there exists a supermartingale U of class (D) on [0, T ] such that E f α,τ ∧σ (Z ρ,δ,γ k ) ≤ U α , α ∈ T 0,τ ∧σ . By replacing γ k by γ k ∧ τ k we may assume that γ k ≤ τ k . By (6.18), (6.19), (H2), (H3) and the Lebesgue dominated convergence theorem, sup ρ,δ∈Sα,γ k E γ k α |f − f n,mn |(r, E f r,τ ∧σ (Z ρ,δ,γ k )) dr → 0 as n → ∞. Thus Y α = Y (α), α ∈ T . By Theorem 6.2, Y is the first component of the solution to RBSDE γ k (Y γ k , f, L, U ). What is left is to show that Y Ta → ∞ as a → ∞. is a solution to RBSDE T (ξ, f n , U m ). By Proposition 6.6(ii), Y n,m ր Y m , and by Propo- Proof. By (H4) and the assumption that |V | T < ∞ a.s., there exists a chain {τ k } on [0, T ] such that Therefore repeating the proof of Theorem 6.7 with γ k replaced by τ k we get the existence of a regulated process Y of class (D) on [0, T ] such that Y is the first component of the solution to RBSDE T (Y τ k , f, L, U ) for every k ≥ 1. It remains to show that Y Ta → ξ as a → ∞. We can not argue as in the proof of Theorem 6.7, because in general, under the assumptions of our theorem, there are no solutions to RBSDE T (ξ, f, U ) and RBSDE T (ξ, f, L). Instead, to show that Y Ta → ξ we use the fact that {τ k } is a chain. By the definition of a solution to RBSDE T (Y τ k , f, L, U ), Y τ k ∧a → Y τ k as a → ∞ for every k ≥ 1. Since {τ k } is a chain, Y τ k (ω) (ω) = Y T (ω) (ω) = ξ(ω), k ≥ k ω , which implies the desired convergence.