Nonlinear BSDEs with Two Optional Doob’s Class Barriers Satisfying Weak Mokobodzki’s Condition and Extended Dynkin Games

We study reflected backward stochastic differential equations (RBSDEs) on the probability space equipped with a Brownian motion. The main novelty of the paper lies in the fact that we consider the following weak assumptions on the data: barriers are optional of class (D) satisfying weak Mokobodzki’s condition, generator is continuous and non-increasing with respect to the value-variable (no restrictions on the growth) and Lipschitz continuous with respect to the control-variable, and the terminal condition and the generator at zero are supposed to be merely integrable. We prove that under these conditions on the data there exists a solution to corresponding RBSDE. In the second part of the paper, we apply the theory of RBSDEs to solve basic problems in Dynkin games driven by nonlinear expectation based on the generator mentioned above. We prove that the main component of a solution to RBSDE represents the value process in corresponding extended nonlinear Dynkin game. Moreover, we provide sufficient conditions on the barriers guaranteeing the existence of the value for nonlinear Dynkin games and the existence of a saddle point.


Introduction
Let B be a standard d-dimensional Brownian motion on a given probability space (Ω, F, P ), T be a strictly positive real number (horizon time) and let F ∶= (F t ) 0≤t≤T be the standard augmentation of the filtration generated by B. In the present paper, we study Reflected Backward Stochastic Differential Equations (RBSDEs for short) of the following form where ξ (terminal value) is an F T -measurable random variable, the mapping generator) is an F-progressively measurable process with respect to the first two variables and L, U (barries) are F-optional processes of class (D).We look for a triple (Y, Z, R) of F-progressively measurable processes, with R of finite variation and R 0 = 0, that satisfies (1.1).Given a solution (Y, Z, R) to (1.1), we call the process Y the main part of the solution.The role of R is to keep Y between barriers L, U , and the role of Z is to keep Y adapted to F. In order to get the uniqueness for problem (1.1) one requires from R to satisfy the so called minimality condition which stands that T 0 where R * is the càdlàg part of R and R * ,+ , R * ,− its Jordan decomposition.
Moreover, it is unique provided Y is of class (D) and Z ∈ H s F (0, T ) -the class of Fprogressively measurable processes satisfying E[ ∫ T 0 Z r 2 dr] s 2 < ∞ -for some s > κ.In the present paper, we focus on the existence and uniqueness problem for (1.1)-(1.2) under conditions (A1)-(A4),(Z).We shall also study the representation of the process Y as the value process in nonlinear Dynkin games.
The existence problem.First, observe that by the very definition of a solution to (1.1) its main part is a semimartingale.Consequently, we deduce at once, that the existence of a semimartingale between the barriers L, U is a necessary condition for the existence of a solution to (1.1) (intrinsic condition).The said condition is known in the literature as weak Mokobodzki's condition (see [19]): (WM) there exists a semimartingale X such that L t ≤ X t ≤ U t , t ∈ [0, T ].
It is a natural question whether under (A1)-(A4), (Z) the above condition is also sufficient for the existence of a solution to (1.1)-(1.2).We give a positive answer to this question, and prove even more, that condition (Z) can be dropped.It appears, and it may seem surprising at first, that the above result does not hold for BSDEs (1.3) (see Remark 3.5).The explanation of this phenomenon is that in the case of reflected BSDEs barriers keep the main part of a solution in the class (D).At this point it is worth mentioning that the following condition (complete separation) L, U are càdlàg, L t < U t , t ∈ [0, T ], L t− < U t− , t ∈ (0, T ] (1.4) implies (WM) (see [36,Lemma 3.1]).We provide a generalization of this condition, by dropping càdlàg regularity assumption on L, U , and we prove that the following condition implies (WM): The uniqueness problem.An interesting issue is also the problem of the uniqueness for solutions to (1.1)-(1.2).In the proof of the uniqueness for BSDEs (1.3) (see [4] and Theorem 3.6) the crucial roles were played by condition (Z) and the fact that for any solution (Y, Z) to (1.3) we have, under conditions (A1)-(A4),(Z), that Z ∈ H s F (0, T ) for some s > κ provided Y is of class (D).For reflected BSDEs this property does not hold even if f ≡ 0 (see [24,Example 5.6]).Nevertheless, we are able to prove the following result.

Theorem 2. Assume that (A1)-(A4),(Z) are in force. Then there exists at most one solution to RBSDE (1.1)-(1.2).
Solutions to RBSDEs as value processes in Dynkin games.The above theorem is a consequence of a much deeper result, which is our third main result of the paper.In order to formulate it, we use the notion of the nonlinear expectation introduced by Peng in [35].For given stopping times α ≤ β ≤ T consider mapping 3), with T replaced by β, such that Y β is of class (D).For given stopping times τ, σ ≤ T and sets H ∈ F τ , G ∈ F σ , we let with the convention that L t = L t∧T , U t ∶= U t∧T , t ≥ 0. We prove the following representation theorem.
for any stopping time θ ≤ T .
In other words, we show that Y is a value process in an extended nonlinear Dynkin game; "nonlinear" since we consider the nonlinear expectation, and "extended" since players may change payoffs L, U on sets H c , G c , respectively, which extends the set of their strategies (in the classical Dynkin games the players are not allowed to choose sets G, H).Observe that the above extended nonlinear Dynkin game reduces to the nonlinear Dynkin game provided L and U are right-continuous.We prove however a stronger result.
for any stopping time θ ≤ T .
Thus, Y represents the value process in a nonlinear Dynkin game provided L, U are sufficiently regular as mentioned above.The above result was achieved by Bayraktar and Yao in [3] for continuous barriers L, U satisfying (1.4) and under the following additional conditions: E sup t≤T L t + E sup t≤T U t < ∞, generator f admits the linear growth with respect to Y -variable, i.e. f (t, y, 0) ≤ g t + ψ y for some ψ ≥ 0. Note that in the present paper growth of f with respect to Y -variable is subject to no restriction.
Finally, we show that further regularity assumptions on barriers L, U allow one to indicate saddle points for nonlinear Dynkin games.For any stopping time θ ≤ T set: Theorem 5. Assume that (A1)-(A4), (Z) are in force.Moreover, suppose that L is upper semicontinuous and U is lower semicontinuous.Then for any stopping time θ ≤ T .
Proof techniques and relations of main results to the existing literature.First, note that Mokobodzki's condition, additionally to (WM), requires from the semimartingale X, lying between the barriers, some integrability of its finite variation and martingale part (depending on the authors you may find different integrability conditions for the process X, nonetheless it is always assumed to be at least the difference of positive supermartingales).This additional requirement is the reason why the complete separation condition (1.4) does not imply Mokobodzki's condition.The fact that (1.4) implies weak Mokobodzki's condition is an easy calculation and may be found e.g. in [36] and [19] (in the case of continuous barriers).Reflected BSDEs with condition (1.4) imposed on the barriers have been considered in many papers (see [3,5,12,18,20,21,22,36]).In the papers [5], [18] and [20], L 2 -data and sublinear growth of the generator with respect to Y -variable are required.In [21] authors considered bounded data and continuous generator with quadratic-growth with respect to Z-variable.Some results for RBSDEs with generators subject to sublinear growth with respect to Y -variable are described also in [3,22] (L 1 -data) and [12] (L p -data, p ∈ (1, 2)).L 1 -data and generator being merely monotone and continuous with respect to Y -variable were considered in [36].
In all the mentioned papers (besides [21]), the method of local solutions and pasting local solutions, introduced by Hamadène and Hassani in [18], has been applied to achieve the existence for underlying RBSDEs.This method is rather complicated, and this is perhaps the reason why the development of theory of RBSDEs with barriers satisfying complete separation condition is far from being satisfactory.The second drawback of the method is that it is based on the penalization scheme which is not available for RBSDEs with optional barriers.In [23], the author proposed a different method which applies to RBSDEs with barriers satisfying even more general than (1.4) weak Mokobodzki's condition (WM).We call this method localization procedure.The advantage of the method is its simplicity and wide applicability.The method is based on the following simple observation: for any chain (τ k ), i.e. non-decreasing sequence of stopping times satisfying The method consists of finding a proper regular approximation (Y n ), on the whole interval [0, T ], of a potential solution Y of a given problem (by "proper" we mean an approximation which does not blow up when passing to the limit).The terms of approximating sequence may solve BSDEs or RBSDEs of the generic form with suitable chosen ξ n , f n , R n .In the first step one shows that (Y n ) converges to a process Y .After that, we show that Y is the main part of a solution to RBSDE τ k (Y τ k , f, L, U ) for each k ≥ 1.Since (τ k ) is a chain, we conclude that Y is the main part of a solution to RBSDE T (ξ, f, L, U ) .
By using localization procedure in [23], the first author of the present paper was able to provide an existence result for RBSDEs with merely càdlàg barriers of class (D) satisfying (WM), L 1 -data, and generator being continuous and non-increasing with respect to Yvariable (with no restrictions on the growth of the generator with respect to Y -variable).
As far as we know the only papers in the literature concerned with RBSDEs of the form (1.1) with non-càdlàg barriers satisfying (WM) are [26,32].In [26] RBSDEs on a general filtered space are studied under (WM) but with f independent of Z-variable.In The generalization of (1.4) to the case of làdlàg barriers was presented in [32] the authors considered làdlàg barriers and stochastic Lipschitz generator f (on the Brownian-Poisson filtration).
As to the nonlinear Dynkin games, to the best of our knowledge, there are only few papers in the literature: [9,10,11,15,17] -all with L 2 -data and Lipschitz generator -and [25,26] -with L 1 -data and continuous and monotone generator with respect to Y -variable and independent of Z-variable.
Comments on the related literature.Reflected backward stochastic differential equations with two barriers have been introduced by Cvitanić and Karatzas in [7] as a generalization of backward stochastic differential equations introduced by Pardoux and Peng in [33] (analogous results for one reflecting barrier, i.e. in case U ≡ ∞, have been presented for the first time by El Karoui et al. in [13]).In [7] the authors considered (1.1) with barriers being continuous processes satisfying Mokobodzki's condition i.e. there exists Moreover, they assumed that data are L 2 -integrable (i.e.sup t≤T L t , sup t≤T U t , ξ , ∫ T 0 f (r, 0, 0) dr have second moments) and f is Lipschitz continuous with respect to (Y, Z)-variable (uniformly in (ω, t)).Under these assumptions a solution to (1.1) has been defined in [7] as a triple (Y, Z, R) of F-progressively measurable processes such that Y is continuous, and R is a continuous finite variation process, with R 0 = 0, satisfying the minimality condition of the form Observe that with continuous Y, L, U, R, condition (1.2) reduces to the above condition.
BSDEs and Reflected BSDEs are of great interest to scientists because of their numerous applications in various fields of mathematics and problems (e.g.partial differential equations, integro-differential equations, variational inequalities, optimization theory, control theory, mathematical finance etc., see [6,34,37] and the references therein).Over the past two decades, many interesting results have been obtained regarding RSBDEs.In particular, numerous existence results for RBSDEs, which strengthen the result of [7] by weakening assumptions on generator f , filtration F, barriers L, U and horizon time T , have been provided.
Despite of intensive research, until 2016, only RBSDEs with càdlàg barriers were considered in the literature.With the work by Grigorova et al. in [14] there was a change in this regard and papers on less regular barriers began to appear.Equations of that type with L 2 -data and Lipschitz generator were studied in [31] (Brownian filtration), in [14,15] (Brownian-Poisson filtration) and [1,2,16] (general filtration).RBSDEs with optional barriers and L 1 -data were considered only in [27,28], in the case of Brownian filtration.Results on optional barriers, L 1 -data and possibly infinite horizon time were presented in [26] but with f independent of Z-variable.The case of L 2 -data and f being stochastic Lipschitz driver was presented in [30] (Brownian-Poisson filtration) and in [29,32] (general filtration).

Basic notation
We say that a function y ∶ [0, T ] → R d is regulated on [0, T ] if for any t ∈ [0, T ), there exists the limit y t+ ∶= lim u↓t y u and for any s ∈ (0, T ] there exists the limit y s− ∶= lim u↑s y u .For any regulated function y on [0, T ] we define ∆ + y t ∶= y t+ −y t , t ∈ [0, T ) and ∆ − y s ∶= y s −y s− , s ∈ (0, T ].
For x ∈ R d by x we denote the euclidean norm.As mentioned in Section 1, T stands for the set of all stopping times taking values in By H F (α, β), we denote the space of all F-progessively measurable, R d -valued processes We say that F-progessively measurable process Throughout the paper all relations between random variables are supposed to hold P -a.s.For processes

Then the following assertions hold. (i) There exists a solution
F (0, T ).Proposition 3.4.Assume that (H5), with p > 1, and (H1),(H2) are satisfied.Let (Y, Z) be a solution to BSDE T (ξ, f ) such that Y ∈ S p F (0, T ).Then there exists c > 0, depending only on µ, λ, T, p, such that In case (H5) is satisfied with p = 1, we shall need for the existence and uniqueness of solutions to BSDEs additional hypothesis.
Observe that, in general, under merely (H1)-( H4) and (H5) with p = 1, we cannot expect the existence of a solution (Y, Z) to BSDE T (ξ, f ) with positive ξ such that Y is positive and of class (D).Indeed, assume that (Y, Z) is a solution to the following BSDE Then, by Itô's formula Therefore, by applying Fatou's lemma, we find that If the above inequality was true for any positive ξ ∈ L 1 (F T ), then exp(B T ) would be bounded, a contradiction.

Reflected BSDEs with two optional barriers under Mokobodzki's condition
In this section we assume that processes L and U are merely where In what follows we refer to condition (b) as the minimality condition.
We consider the following condition, which we call strong Mokobodzki's condition:

(i) There exists at most one solution
In the case of p = 1, we consider the following version of strong Mokobodzki's condition: The following result has been proven in [28,Theorem 3.8].

Nonlinear expectation
Let p ≥ 1.Throughout this section, we assume that either p = 1 and (H1)-(H5), (Z) are in force or p > 1 and (H1)-(H5) are in force.Let α, β ∈ T , α ≤ β.We define the operator . By Theorem 3.6, the operator E (1),f α,β is well defined under conditions (H1)-(H5), (Z).By Theorem 3.3, under (H1)-(H5) (with p > 1), we may define the operator F (0, β).Finally, we define operator We say that a process X of class (D) is an (i) Let ξ ∈ L p (F β ) and let V be an F-adapted, finite variation process such that V α = 0. Let (X, H) be a solution to BSDE α,β (ξ, f + dV ) such that X is of class (D), in case p = 1, and where for some C depending only on λ, µ, T .(vi) Let p > 1. Assume that f 1 , f 2 satisfies (H1)-(H5) and let α, Then there exists c > 0, depending only on T, µ, λ, p, such that such that such that X is of class (D), in case p = 1, and X ∈ S p F (α, τ ), in case p > 1.By [28, Proposition 3.2, Lemma 3.3] and Theorem This completes the proof of (i).The assertion (ii) follows directly from [28, Proposition 3.2, Lemma 3.3] and Theorem 3.6.As to (iii), let By the uniqueness for BSDEs (see Theorems 3.3,3.6) which implies (iii).For (iv), let (Y, Z) be a solution to BSDE β (ξ, f ) and let ( Ȳ , Z) be a solution to This concludes the proof of (iv).Now, we shall proceed to the proof of (v).Let (Y 1 , Z 1 ), (Y 2 , Z 2 ) be defined as in the assertion (v).By (iv), we know that From the definition of a solution to BSDE, we have that {τ k } k≥1 is a chain.By Ito's formula, (5.2) By (H1), (H2) (without loss of generality we may assume that µ = 0), we have ).Therefore, by (5.2), we have Consequently, by the fact that , we may conclude, by letting k → ∞ in the right-hand side of (5.3) and applying the Lebesgue dominated convergence theorem, that for some C > 0 depending only on λ and β 2 .Finally, note that which combined with (5.4) completes the proof of (v).The inequality asserted in (vi) follows directly from Proposition 3.4.

Extended nonlinear Dynkin games
In the whole section, we assume that (H1)-(H5), (Z) are in force and that L and U are F-optional processes of class (D).Definition 6.1.Let τ ∈ T and H ∈ F τ .A pair ρ = (τ, H) is called a stopping system if {τ = T } ⊂ H.For brevity, we write τ ⌊H.
By U we denote the set of all stoping systems.We then have T ⊂ U, by using embedding T ∋ τ ↦ τ ⌊Ω ∈ U.For given θ ∈ T , we denote by U θ the set of all stopping systems τ ⌊H such that τ ≥ θ.For an optional, right-limited process φ and τ ⌊H ∈ U we put In particular, we have φ τ ⌊Ω = φ τ .For an optional process φ we let Note that, when φ is right-limited, then φ u τ ⌊H = φ l τ ⌊H = φ τ ⌊H .For two stopping systems τ ⌊H, σ⌊G ∈ U we define the pay-off Note that J(τ ⌊H, σ⌊G) is F τ ∧σ -measurable random variable.Now, we shall proceed to the so called extended Dynkin games.Definition 6.2.Let θ ∈ T .
(i) Upper and lower value of the game are defined respectively as (ii) We say that an extended E f -Dynkin game with pay-off function J has a value if Let (Y, Z, R) be a solution to RBSDE T (ξ, f, L, U ).For every θ ∈ T and ε > 0 we define the following sets Let us also define the following stopping times Consider the following stopping systems Thus, by Proposition 5. (6.5)

Proof. (i)
We shall prove the first inequality in (6.4), the proof of the second one runs analogously.Due to the definitions of On the other hand, by the definition of τ ε θ , for P-a.e. ω ∈ Ω there exists nonincreasing sequence Due to the definiton of (ii) First, we shall prove the first inequality in (6.5).We have By the form of H ε and the minimality condition, By virtue of (6.7)-(6.9),we conclude that The proof of the second inequality in (6.5) requires slightly different arguments.We have (6.10) By the analogous argument as in the proof of (6.7) -the form of G ε combined with the definition of a solution to RBSDE give But we also need . Therefore, (6.13) must hold.Combining (6.10)- (6.13) gives we easily deduce from the above inequality the result.Lemma 6.5.Let (Y, Z, R) be a solution to RBSDE T (ξ, f, L, U ).We have the following inequalities: where C is a constant depending only on λ, µ, κ, T, g L 1 , γ.
Proof.Let θ ∈ T and ε > 0. We shall show the first inequality in (6.14) (the proof of the other one is analogous).By Lemma 6.4 We have By (6.15) and by properties of the operator E f (see Proposition 5.2 (ii) and (v)), we get The extended E f -Dynkin game has a value.What is more, for any stopping time θ ∈ T , Moreover, for every θ ∈ T and ε > 0 the pair of stopping systems (τ ε θ ⌊H ε , δ ε θ ⌊G ε ) defined in (6.3) is ε-saddle point in time θ for extended E f -Dynkin game, i.e. satisfies inequalities (6.4).
Proof.Since right-hand side inequality in (6.15) is satisfied for all σ⌊B ∈ U θ we have that Thus, by the definition of V θ (see (6.2)) we have that

Nonlinear Dynkin games
Throughout the section, we assume that (H1)-(H5), (Z) are in force and that L and U are F-optional processes of class (D).
For τ, σ ∈ T we define the pay-off Assume that L is right upper semicontinuous and U is right lower semicontinuous.Then Without loss of generality we may assume that properties attributed to Y, M, R, L, U and holding P -a.s.hold for any ω Take ω ∈ Ω.By the definition of τ ε θ there exists a non-increasing sequence (ω) + ε, which contradicts (7.4).From this we deduce that Theorem 7.3.Assume that L is right upper semicontinuous and U is right lower semicontinuous.Let (Y, Z, R) be a solution to RBSDE T (ξ, f, L, U ). Then for any θ ∈ T ) where C is a constant depending only on λ, µ, α, T, g L 1 , γ.
Proof.Let θ ∈ T and ε > 0. We shall prove that (τ ε θ , σ ε θ ) satisfies (7.6).By Lemma 6.3 We thus have By the assumptions made on L and Lemma 7.2, Y τ ε θ ≤ L τ ε θ + ε.From this and the fact that Y ≤ U we have Applying (7.7) and properties of the operator E f (see Proposition 5.2 (ii) and (v)) yields By the assumptions made on U and Lemma 7.2 we have − Cε, which combined with (7.8) gives (7.6).Consequently, 8. Existence of saddle points.
In the whole section, we assume that (H1)-(H5), (Z) are in force and that L and U are F-optional processes of class (D).
Let (Y, Z, R) be a solution to RBSDE T (ξ, f, L, U ).We shall prove that there exists a saddle point for a nonlinear Dynkin game with sufficiently regular payoffs.For θ ∈ T we define: and Proof.Ad 1).Assume that R −, * is continuous.By the definition of σθ we have that R − σθ = R − θ .Thus, for any a ≥ 0, We shall prove that Y σθ = U ξ σθ .Assume that σθ < T (in case σθ = T the desired equality is obvious).Suppose, by contradiction, that P (Y σθ < U ξ σθ ) > 0. By the minimality condition, This contradicts the minimality condition.Proof.Let τ ∈ T be predictable.We shall prove that ∆ − R +, * τ = 0. We have where D ∶= {∆R +, * τ > 0} and D ′ ∶= {∆R −, * τ > 0}.Since dR + ⊥ dR − , D ∩ D ′ = ∅.Thus, on the set D, ∆Y τ ≤ 0. From this and the regularity assumption on L, Since the last inequality holds for any predictable τ ∈ T , we deduce that R +, * is continuous.
The similar reasoning may be applied to U .
Since Y ≥ L and Y σ * θ = U σ * θ (see Proposition 8.1), we also have Using (8.6) and the fact that E f is a non-decreasing operator, we deduce that In the similar way we arrive at ) is a saddle point at θ. Analogously, one shows, by using Proposition 8.1, that (τ θ , σθ ) is a saddle point at θ.

Existence result
In the whole section, we assume that L, U are F-optional processes of class (D).
Let us consider the following assumption, which is called in the literature weak Mokobodzki's condition.
(WM) There exists a semimartingale X such that L ≤ X ≤ U .Proposition 9.1.Assume that L, U are left-limited, and Then weak Mokobodzki's condition (WM) holds for L, U .
Step 2. We shall construct a semimartingale lying between barriers L, U .Define get that X is of finite variation, thus a semimartingale.This completes the proof.
Proof.Let X be the process appearing in (H7).Since X is a special semimartingale, there exists a chain (γ k ) and processes H ∈ H F (0, T ) and Consequently, by the Lebesgue dominated convergence theorem, the most right term in (9.9) tends to zero as n → ∞.As a result, letting n → ∞ in (9.9), we obtain that Y n,m → Ỹ k,m in D 2 F (0, γ k ).This completes the proof of step 1.

Definition 3 . 1 .
and ξ ∈ F β .We say that a pair (Y, Z) of F-adapted processes is a solution to backward stochastic differential equation on the interval [[α, β]] with right-hand side f and terminal value

Definition 3 . 2 .
We say that a pair (Y, Z) of F-adapted processes is a solution to backward stochastic differential equation on the interval [[α, β]] with right-hand side f + dV and terminal value ξ

Definition 4 . 1 .
We say that a triple (Y, Z, R) of F-adapted processes is a solution to reflected backward stochastic differential equation on the interval [[α, β]] with right-hand side f , terminal value ξ, lower barrier L and upper barrier U

. 1 ) 7 . 1 .
Definition Let θ ∈ T .Upper and lower value of the game are defined respectively as