A general Doob-Meyer-Mertens decomposition for $g$-supermartingale systems

We provide a general Doob-Meyer decomposition for $g$-supermartingale systems, which does not require any right-continuity on the system. In particular, it generalizes the Doob-Meyer decomposition of Mertens (1972) for classical supermartingales, as well as Peng's (1999) version for right-continuous $g$-supermartingales. As examples of application, we prove an optional decomposition theorem for $g$-supermartingale systems, and also obtain a general version of the well-known dual formation for BSDEs with constraint on the gains-process, using very simple arguments.


Introduction
The Doob-Meyer decomposition is one of the fundamental result of the general theory of processes, in particular when applied to the theory of optimal control, * We are very grateful to N. El Karoui for discussions we had on the first version of this paper.
† CEREMADE and CREST-LFA, University Paris-Dauphine and ENSAE-ParisTech. ANR Liquirisk. bouchard@ceremade.dauphine.fr ‡ CEREMADE, University Paris-Dauphine. possamai@ceremade.dauphine.fr § CEREMADE, University Paris-Dauphine. tan@ceremade.dauphine.fr, the author gratefully acknowledges the financial support of the ERC 321111 Rofirm, the ANR Isotace, and the Chairs Financial Risks (Risk Foundation, sponsored by Société Générale) and Finance and Sustainable Development (IEF sponsored by EDF and CA). see El Karoui [17]. Recently, it has been pointed out by Peng [40] that it also holds in the semi-linear context of the so-called g-expectations. Namely, let (Ω, F, P) be a probability space equipped with a d-dimensional Brownian motion W , as well as the Brownian filtration F = (F t ) t≥0 , let g : (t, ω, y, z) ∈ R + ×Ω×R×R d −→ R be some function, progressively measurable in (t, ω) and Lipschitz in (y, z), and ξ ∈ L 2 (F τ ) for some stopping time τ . We define E g ·,τ [ξ] := Y · where (Y, Z) solves the backward stochastic differential equation with terminal condition Y τ = ξ. Then, an optional process X is said to be a (strong) E g -supermartingale if for all stopping times σ ≤ τ we have X τ ∈ L 2 (F τ ) and X σ ≥ E g σ,τ [X τ ] almost surely. When X is right-continuous, it admits a unique decomposition of the form in which Z X is a square integrable and predictable process, and A X is nondecreasing predictable. See [40] and [10,25,32]. In particular, when g ≡ 0, this is the classical Doob-Meyer decomposition in a Brownian filtration framework.
However, it is limited to right-continuous E g -supermartingales, while the rightcontinuity might be very difficult to prove, if even correct. The method generally used by the authors is then to work with the right-limit process, which is automatically right-continuous, but they then face important difficulties in trying to prove that it still shares the dynamic programming principle of the original process. This was sometimes overcome to the price of stringent assumptions, which are often too restrictive, in particular in the context of singular optimal control problems.
In the classical case, g ≡ 0, it is well known that we can avoid these technical difficulties by appealing to the version of the Doob-Meyer decomposition for supermartingales with only right and left limits, see El Karoui [17]. It has been established by Mertens [35], Dellacherie and Meyer [12, Vol. II, Appendice 1] provides an alternative proof. Unfortunately, such a result has not been available so far in the semi-linear context. This paper fills this gap 1 and provides a version à la Mertens of the Doob-Meyer decomposition of E g -supermartingales. By following the arguments of Mertens [35], we first show that a supermartingale associated to a general family of semi-linear (non-expansive) and time consistent expectation operators can be corrected into a right-continuous one by subtracting the sum of the previous jumps on the right. Applying this result to the g-expectation context, together with the decomposition of [40], we then obtain a decomposition for the original E g -supermartingale, even when it is not right-continuous. The same arguments apply to g-expectations defined on L p , p > 1, and more general filtrations than the Brownian one considered in [40], in particular we shall not assume that the filtration is quasi left-continuous. This is our Theorem 3.1 below. The only additional difficulty is that the decomposition for right-continuous processes has to be extended first. This is done by using the fact that it can naturally be obtained by considering the BSDE reflected from below on the E g -supermartingale and by using recent technical extensions of the seminal paper El Karoui et al. [19], see Proposition 3.1 below. Then, using classical results of the general theory of stochastic processes, we can even replace the notion of supermartingale by that of supermartingale systems, for which an optional aggregation process can be easily found, see El Karoui [17] for the classical case g ≡ 0.
These key statements aim not only at extending already known results to much more general contexts, but also at simplifying many difficult arguments recently encountered in the literature. We provide two illustrative examples. First, we prove a general optional decomposition theorem for g-supermartingales. To the best of our knowledge, such a decomposition was not obtained before. Then, we show how a general duality for the minimal super-solution of a backward stochastic differential equation with constraint on the gains-process can be obtained. This is an hold problem, but we obtain it in a framework that could not be considered in the literature before, compare with [2,11]. In both cases, these a-priori difficult results turn out to be easy consequences of our main Theorem 3.1, whenever right continuity per se is irrelevant.
Notations: (i) In this paper, (Ω, F, P) is a complete probability space, endowed with a filtration F = (F t ) t≥0 satisfying the usual conditions. Note that we do not assume that the filtration is quasi left-continuous.
(ii) We fix a fixed time horizon T > 0 throughout the paper, and denote by T the set of stopping times a.s. less than T . We shall also make use of the set T σ of stopping times τ ∈ T a.s. greater than σ ∈ T . For ease of notations, let us say that (σ, τ ) ∈ T 2 if σ ∈ T and τ ∈ T σ .
(iii) Let σ ∈ T , conditional expectations or probabilities given F σ are simply denoted by E σ and P σ . Inequalities between random variable are taken in the a.s. sense unless something else is specified. If Q is another probability measure on (Ω, F), which is equivalent to P, we will write Q ∼ P.
(iv) For any sub-σ-field G of F, L 0 (G) denotes the set of random variables on (Ω, F) which are in addition G-measurable. Similarly, for any p ∈ (0, ∞], and any probability measure Q on (Ω, F), we let L p (G, Q) be the collection of real-valued G-measurable random variables with absolute value admitting a p-moment under Q. For ease of notations, we denote L p (G) := L p (G, P) and also L p := L p (F). These spaces are endowed with their usual norm.
(v) For p ∈ (0, ∞], we denote by X p (resp. X p r , X p ℓr ) the collection of all optional processes X such that X τ lies in L p (F τ ) for all τ ∈ T (resp. and such that X admits right-limits, and such that X admits right-and left-limits). We denote by S p the set of all càdlàg, F-optional processes Y , such that sup 0≤t≤T Y t ∈ L p , and by H p the set of all predictable d-dimensional processes Z such that Finally, we denote by A p the set of all non-decreasing predictable processes A such that A 0 = 0 and A T ∈ L p .
(vi) For any d ∈ N\{0}, we will denote by x · y the usual inner product of two elements (x, y) ∈ R d × R d . We will also abuse notation and let |x| denote the Euclidean norm of any x ∈ R d , as well the associated operator norm of any d × d matrix with real entries.

Stability of E-supermatingales under Mertens's regularization
In this section, we provide an abstract regularization result for supermartingales associated to a family of semi-linear non-expansive and time consistent conditional expectation operators (see below for the exact meaning we give to this, for the moment, vague appellation). It states that we can always modify a supermartingale with right-limits so as to obtain a right-continuous process which is still a supermatingale. This was the starting point of Mertens's proof of the Doob-Meyer decomposition theorem for supermatingales (in the classical sense) with only right-limits. Our proof actually mimics the one of Mertens [35]. This abstract formulation has the merit to point out the key ingredients that are required for it to go through, in a non-linear context. It will then be applied to g-expectation operators, in the terminology of Peng [39], to obtain our Doob-Meyer type decomposition, which is the main result of this paper.

Semi-linear time consistent expectation operators
Let p ∈ (1, +∞]. Throughout the paper, q will denote the conjugate of p (i.e. p −1 + q −1 = 1). Then, we define a non-linear conditional expectation operator as a family E = {E σ,τ , (σ, τ ) ∈ T 2 } of maps One needs it to satisfy certain structural and regularity properties. Let us start with the notions related to time consistency.
We also need some regularity with respect to monotone convergence.
(b) Let (σ n ) n≥1 ⊂ T be a decreasing sequence which converges a.s. to σ and s.t. σ n ≤ τ a.s. for all n ≥ 1. Fix ξ ∈ L p (F τ ). Then, (c) Let (ξ n ) n≥1 ⊂ L p (F τ ) be a non-decreasing sequence which converges a.s. to ξ ∈ L p . Then, lim sup The idea that E should be semi-linear and non-expansive is encoded in the following. Let Q 1 , Q 2 be two probability measures on (Ω, F) and τ ∈ T , we define the concatenated probability measure Q 1 ⊗ τ Q 2 on (Ω, F) by Assumption (Sld). There is a family Q of P-equivalent probability measures such that: Let us comment this last condition. Assume that (Q, β) is the same for (ξ, ξ ′ ) and (ξ ′ , ξ). Then, inverting the roles of ξ and ξ ′ , it indeed says that . Otherwise stated, in this case, the operator E can be linearized as each point. However, the linearization, namely (Q, β), depends in general on (ξ, ξ ′ ), σ and τ , so that it is not a linear operator. Thus the label semi-linear.

Stability by regularization on the right
Before stating the main result of this section, one needs to define the notion of E-supermartingales.
Define the process I by Then, I is non-decreasing, left-continuous and belongs to X 1 p . Moreover, X := X + I is a right-continuous local E-supermatingale.
Proof. We split the proof in several steps. As already mentioned, we basically only check that the arguments of Mertens [35] go through under our assumptions.
(a) X is right-continuous. Indeed, for every t ∈ [0, T ), one has (b) Jumps from the right are non-positive, i.e. X t ≥ X t+ for each t ∈ [0, T ), so that I is non-decreasing, and (c) Let k ∈ N, ε > 0, and (σ i ) i≤k ⊂ T be the non-decreasing sequence of stopping times which exhausts the first k jumps from the right of X of size bigger than ε (recall that X admits right-limits). Denote Note that we can always assume that there is a.s. at least k jumps, as we can always add jumps of size 0 at T . We shall use the conventions σ 0 = 0 and σ k+1 = T . The proof proceeds by induction and requires several steps. For ease of notation, we omit the superscript (ε, k) in (X ε,k , I ε,k ) and write (X, I) in this part (that is in item (c) only).
Indeed, since X ≥ X + and I ≥ 0 by (b), (Sld) implies that On the other hand, it follows from (ii) In view of (Tc)(b), the result of (i) implies in particular that (iii) Given τ ∈ T , we next show by induction that For i = k, this follows from (Tc)(c) and (i). Assume that it is true for But, by (a) and (c) of (Tc) and the induction hypothesis, we deduce immediately (iv) We are in position to conclude this step.
But it follows from (iii), and the same arguments as above, that Recalling the result of (i), we conclude that, on this concludes the proof of this step. (d) We now provide a bound on I ε,k T defined by (2). Let (σ i ) i≤k be as in (c) associated to the parameter (ε, k). We first prove by induction that The result is true for i = k + 1, recall our convention σ k+1 = T and (Tc)(a). Let us assume that it holds for some i + 1 ≤ k + 1. Then, by (Tc)(a)-(b) and (Sld) combined with (b), in whichQ i ∈ Q. Then, our induction claim follows for i by composingQ i and Q i+1 in an obvious way. Recalling our convention σ 0 = 0, this implies that Since p and q are conjugate, it remains to use Hölder's inequality to deduce that for some C L > 0 which only depends on L.
(e) We now extend the bound (3) to the general case.
Notice that the r.h.s. of (3) does not depend on ε nor k, so we can first send k to ∞ and then ε to 0 and apply the monotone convergence theorem, to obtain that (f) It remains to show that X := X + I is a local E-supermartingale.
Recall that I is defined in (1), and (I ε,k , X ε,k ) are defined in (2). Let ϑ n be the first time when I ≥ n. Note that (ϑ n ) n≥1 is a.s. increasing and converges to ∞, this follows from (e). We know from (c) that X ε,k is a E-supermartingale. Hence, But X ε,k ϑ ↑ X ϑ a.s. for any stopping time ϑ, when one let k first go to ∞ and then ε to 0. Since X τ ∧ϑn ∈ L p (F τ ), by definition of (ϑ n ) n≥1 and the fact that which concludes the proof.

Doob-Meyer-Mertens decomposition of g-supermartingale systems
We now specialize to the context of g-expectations introduced by Peng [39] (notice however that we consider a slightly more general version). The object is to provide a Doob-Meyer-Mertens decomposition of g-supermartingale systems without càdlàg conditions. This is our Theorem 3.1 below.
We assume that the space (Ω, F, P) carries a d-dimensional Brownian motion W , adapted to the filtration F, which may be strictly larger than the natural (completed) filtration of W . Recall that F satisfies the usual conditions.

g-expectation and Doob-Meyer decomposition
for some constant number L g > 0. We also assume that (g t (ω, 0, 0)) t≤T satisfies the following integrability condition In the following, we most of the time omit the argument ω in g.
such that (Y, Z) ∈ S p × H p and N is a càdlàg F-martingale orthogonal to W in the sense that the bracket [W, N ] is null, P-a.s., and such that We also remind the reader that the introduction of the orthogonal martingale N in the definition of the solution is necessary, since the martingale predictable representation property may not hold with a general filtration F. The map E g is usually called the g-expectation operator.
We define E g -supermartingales, also called g-supermatingales, as in the previous section, for E = E g , i.e. X is a E g -supermatingale iff X ∈ X p and X σ ≥ E g σ,τ [X τ ] a.s. for all (σ, τ ) ∈ T 2 . For càdlàg E g -supermartingales, we have the following classical Doob-Meyer decomposition, which is a consequence of the well-posedness of a corresponding reflected backward stochastic differential equation. Its proof is provided in the Appendix (see also Peng [40,Thm. 3.3] in the case of a Brownian filtration). 2 Proposition 3.1. Let X ∈ X p be a càdlàg E g -supermartingale. Then there exists Z ∈ H p , a càdlàg process A ∈ A p and a càdlàg martingale N , orthogonal to W , for all (σ, τ ) ∈ T 2 . Moreover, this decomposition is unique.

Time consistence and regularity of g-expectations
We now verify that the conditions of Lemma 2.1 apply to E g . Proposition 3.2. Assume that y −→ g t (ω, y, z) is non-increasing for all z ∈ R, for dt × dP − a.e. (t, ω) ∈ [0, T ] × Ω. Then, Assumptions (Tc), (S) and (Sld) hold for E g .
Proof. First, notice that since W is actually continuous, we not only have [W, N ] = 0, a.s., but also where N c (resp. N d ) is the continuous (resp. purely discontinuous) martingale part of N . Then (Tc) follows from the definition of E g and the uniqueness of a solution. The stability properties (S)(b) and (c) follow from the path continuity of the Y component of the solution of (6) and the standard estimates given in [4, Thm 2.1 and Thm 4.1], see also [30,Prop. 3] for the case where the filtration is quasi left-continuous 3 .
The fact that (Sld) holds is a consequence of the usual linearization argument. Let (Y, Z, N ) and (Y ′ , Z ′ , N ′ ) be the solutions of (6) with terminal conditions ξ and ξ ′ . Then, since g is uniformly Lipschitz continuous, there exist two processes λ and η, which are F-progressively measurable, such that These two processes are bounded by L g for dt × dP − a.e. (t, ω) ∈ [0, T ] × Ω, as a consequence of (4). Moreover, λ ≤ 0 since g is non-increasing in y.
Then, for any 0 ≤ t ≤ s ≤ T , let us define the following continuous, positive and F-progressively measurable process By applying Itô's formula, we deduce classically (see [30,Lem. 9 which is nothing else but Assumption (Sld) by Girsanov's theorem (recall that λ ≤ 0 and that λ and η are bounded by L g , i.e. it suffices to consider Q as the collection of measures with density with respect to P given by an exponential of Doléans-Dade of the above form with η bounded by L g ).
Finally, the condition (S)(a) follows from a similar linearization argument. Let s ∈ [0, T ) and ξ ∈ L 0 (F s ), s n ց s and (ξ n ) n≥1 be such that ξ n ∈ L p (F sn ) for each n, (ξ − n ) n≥1 is bounded in L p and ξ n → ξ as n → ∞. One has for a sequence (A n ) n≥1 bounded in any L p ′ , p ′ ≥ 1, which converges a.s. to 1, and some C ≥ 1 independent on n. Since ξ − n , sn s |g s (0, 0)|ds n≥1 is bounded in L p , and p > 1, the negative part of term in the above expectation is uniformly integrable, and we can apply Fatou's Lemma to conclude the proof.
Remark 3.1. One easily checks that X σ+ ≥ E g σ,τ [X τ + ] for (σ, τ ) ∈ T 2 , whenever X is a E g -supermartingale. Again, this follows from the path continuity of the Y component of the solution of (6) and the estimates of [4, Rem 4.1].
Corollary 3.1. Assume that y −→ g t (ω, y, z) is non-increasing for all z ∈ R d , for dt × dP − a.e. (t, ω) ∈ [0, T ] × Ω. Let X ∈ X p r be an E g -supermartingale. Define the process I by Then, I is a non-decreasing and left-continuous process satisfying I T ∈ L 1 p . Moreover, X := X + I is a right-continuous local E g -supermatingale.
Proof. This is an immediate consequence of Lemma 2.1 and Proposition 3.2 if (X − t ) t≤T is bounded in L p . But this follows from the fact that For later use, let us a provide another version in which the monotonicity of g in y is not used anymore. The price to pay is that the I process defined below may not be non-decreasing anymore.
Corollary 3.2. Let X ∈ X p r be an E g -supermartingale. Then, X t ≥ X t+ for all t ∈ [0, T ). Define the process I by Then, I T ∈ L 1 p , I is left-continuous. Moreover, X := X + I is a right-continuous local E g -supermatingale.
Proof. It follows from Corollary 3.1 that the result holds if g is non-increasing in its y-variable. On the other hand, it is immediate to check that ζ is an E gsupermartingale if and only ifζ is an Eg-supermartingale, with ζ := e LgT ζ andg t (y, z) := e Lgt g(ye −Lg t , ze −Lg t ) − L g y.
The mapg is now non-increasing in its y-component as a consequence of (4). Moreover,g still satisfies (4), with the same constant L g , and, by (5),g(0, 0) satisfies the integrability condition needed to define the corresponding BSDE. HenceX +Ĩ is a right-continuous Eg-supermartingale,Ĩ is non-decreasing and Hence, X +I = e −Lg · (X +Ĩ) is a E g -supermartingale, and I T ∈ L 1 p sinceĨ T ∈ L

The Doob-Meyer-Mertens's decomposition for E g -supermartingales
We are now in position to state the main result of this paper.
Proof. (a) Let us first prove that there exists an optional process X ∈ X p such that S(σ) = X σ a.s. for all σ ∈ T . Since S is uniformly integrable, [13, Thm. 6 and Rem. 7 c)] imply that it suffices to show that for all non-increasing sequence (σ n ) n≥1 ∈ T σ such that σ n −→ σ ∈ T , a.s. By using a similar linearization argument as the one used in the proof of Proposition 3.2, we can find F-progressively measurable processes λ n and η n that are bounded by L g dt × dP-a.e. and such that Note that (H n σn − 1)S(σ n ) ≥ −(S(σ n )) + − H n σn (S(σ n )) − . Since S is uniformly integrable, so is S + . Besides, we have by definition . But, once more it is clear that E g σn,T [S(T )] is bounded in L p , uniformly in n, see [4,Thm 4.1]. Since H n σn has bounded (uniformly in n) moments of any order, de la Vallée-Poussin criterion ensures that H n S − is also uniformly integrable. Therefore, {[(H n σn − 1)S(σ n )] − , n ≥ 1} is uniformly integrable. Using the fact that (λ n , η n ) n is uniformly bounded by L g , as well as (5), we can use Fatou's lemma in (9) to obtain that the second and the third terms on the right-hand side converges to 0 as n −→ ∞.
(b) The fact that X has right-and left-limits, up to an evanescent set, follows from Lemma A.2 stated below, since X is an E g -supermartingale.
(c) Let I be defined as in Corollary 3.2 for X. Since X +I is right-continuous, we can apply the Doob-Meyer decomposition of Proposition 3.1 to X n := (X + I) ·∧ϑn where ϑ n is the first time when I ≥ n. There exists (Z n ,Ā n ) ∈ H p × A p and a càdlàg martingale N n , orthogonal to W , such that, for (σ, τ ) ∈ T 2 , in which η is a progressively measurable process bounded by L g , dt × dP-a.e., as a consequence of (4). Set and observe that (A n , Z n , for n ≤ k, by uniqueness of the decomposition in Proposition 3.1. We can then define so that We claim that A is non-decreasing and that the above decomposition is unique. The fact that (Z, A, [N ] T ) ∈ H p × A p × L p/2 then follows from [4, Proposition 2.1].
Let us now prove our claim. DefineX andg bỹ X := e Lg · X andg t (y, z) := e Lg t g(ye −Lg t , ze −t ) − L g y.
Then,X is a Eg-supermartingale, and so is its right-limits processX + :=X + , as a consequence of Remark 3.1, recall Lemma A.2 below. Applying Proposition 3.1, we can find a right-continuous non-decreasing processÃ ∈ A p ,Z ∈ H p and a càdlàg martingaleÑ , orthogonal to W , such that This decomposition is unique. On the other hand, (12) implies that Hence,B =Ã is non-decreasing. But, since (g(X, e Lg· Z) −g(X + , e Lg · Z)) ≤ 0 as a consequence of Corollary 3.2 (namelyX ≥X + ) and the fact thatg is non-increasing in its first component, we must have that the continuous part of · 0 e Lg s dA s is non-decreasing, and so must be the continuous part of A. We now deduce from the definition of I in (8) and (10)-(11) that A can only decrease in a continuous manner, recall thatĀ n is non-decreasing. Hence, A is non-decreasing. The fact that the decomposition is unique comes from the uniqueness of the decomposition forX + .

Remarks
The framework of this section corresponds to the case where the BSDEs are driven by a continuous martingale M , whose quadratic variation is absolutely continuous with respect to the Lebesgue measure, and with an invertible density. Extensions to the context of [6], see also [18], [29] or [9], would be of interest. Similarly, one could certainly consider BSDEs with jumps, generators with quadratic growth, obstacles, stochastic Lipschitz conditions, etc. We have chosen to work in a simpler setting so as not to drown our arguments with unneeded technicalities, and to focus on the novelty of our approach.
However, the case p = 1 can not be treated by the same technics, in particular we can not appeal to the classical linearization procedure. It would also require a reinforcement of the condition (4), see [5].

Applications
We now consider two problems studied in the recent literature, which are solved with sophisticated arguments under technical conditions. Using Theorem 3.1, we can solve these problems in a very general context with quite simple arguments.

Optional decomposition of g-supermartingale systems
We are still in the context of the previous section, with the slight modification that, instead of the Brownian motion W , we consider a continuous (P, F)martingale M of the form: M t = t 0 α ⊤ s dW s , in which α is a R d×d -bounded predictable process with bounded inverse. Recall that F satisfies the usual conditions.
Let S = {S(τ ), τ ∈ T } be a T -system, g be as in Section 3 such that (4) and (5) hold. Let M 0 denote the set of probability measures Q on (Ω, F) which are equivalent to P and such that M is a (Q, F)-martingale. We then say that a Tsystem S is a E g -supermartingale system under some Q ∈ M 0 if S(τ ) ∈ L p (Q) for all τ ∈ T and S(σ) ≥ E Q,g σ,τ [S(τ )] for all (σ, τ ) ∈ T 2 , where, with (σ, τ ) ∈ T 2 and ξ ∈ L p (F τ , Q), we set E Q,g σ,τ [ξ] := Y σ , with (Y, Z, N ) the unique solution of such that Y ∈ S p (Q), Z belongs to H p (Q) and N is a càdlàg (F, Q)-martingale orthogonal to M , and such that The spaces S p (Q) and H p (Q) are defined as S p and H p , but with Q instead of P.
The main result of this section is the following optional type decomposition (see e.g. [20,26,21]).

Theorem 4.1 (Optional decomposition).
If for any Q ∈ M 0 , S is a E Q,g -supermartingale system which is Q-uniformly integrable s.t. esssup{|S(τ )|, τ ∈ T } ∈ L p (Q), then there exists (X, Z) ∈ X p ℓr × H p such that S(σ) = X σ for all σ ∈ T , and Proof. The existence of the process X ∈ X p ℓr such that S(σ) = X σ for all σ ∈ T follows from Theorem 3.1. Fix then some Q ∈ M 0 . Using Theorem 3.1, we deduce the existence of (Z Q , A Q ) ∈ H p (Q) × A p (Q) and of a Q-martingale N Q orthogonal to M such that P − a.s. (or Q − a.s.) Recall the definition of I in Corollary 3.2 and that X + I is right-continuous. Then, and the family (Z Q ) Q∈M 0 can actually be aggregated into a universal predictable process Z, since α is invertible. Hence, we deduce that X + · 0 g s (X s , Z s )ds is actually a supermartingale under any Q ∈ M 0 , and we can apply the classical optional decomposition theorem ([21, Thm.1]) together with the classical Mertens's decomposition ([35, T2 Lemme]) to deduce the existence of an F-predictable process Z such that Next, using (13), we obtain Z = Z dt × dP-a.e., which ends the proof.

Dual formulation for minimal super-solutions of BSDEs with constraints on the gains process
In this section, we provide an application to the dual representation for BSDEs with constraints. We specialize to the situation where Ω is the canonical space of R d -valued continuous functions on [0, T ], starting at 0, endowed with the Wiener measure P. We let F • = (F • t ) t≤T denote the raw filtration of the canonical process ω −→ W (ω) = ω, while F denotes its P-augmentation. We also fix p ′ > p > 1.
We let g be as in Section 3 such that (4) and (5) hold for p ′ and fix ξ ∈ L p ′ . Further, let O = (O t (ω)) (t,ω)∈[0,T ]×Ω be a family of non-empty closed convex random subsets of R d , such that O is F • -progressively measurable in the sense of random sets (see e.g. Rockafellar [45] In particular, it admits a Castaing representation, see e.g. [45], which in turn ensures that the support function defined by under the constraint We say that a solution (Y, The dual characterization relies on the following construction. Let us also define U as the class of R d -valued, progressively measurable processes such that |ν| + |δ(ν)| ≤ c, dt × dP-a.e., for some c ∈ R. Given ν ∈ U , we let P ν be the probability measure whose density with respect to P is given by the Doléans-Dade exponential of · 0 ν s · dW s , and denote by W ν := W − · 0 ν s ds the corresponding P ν -Brownian motion. Then, given In the above, S p (P ν ) and H p (P ν ) are defined as S p and H p but with respect to P ν in place of P.
Before providing the proof of this result, let us comment it. This formulation is known since [11], however it was proven only under strong assumptions. Although it should essentially be a consequence of the Doob-Meyer decomposition for g-supermatingales, the main difficulty comes from the fact that the family of controls in U is not uniformly bounded. Hence, (16) is a singular control problem for which the right-continuity of τ −→ S(τ ) is very difficult to establish, a priori, see [2] for a restrictive Markovian setting. This fact prevents us to apply the result of [40]. Theorem 3.1 allows us to bypass this issue and provides a very simple proof.

A.1 Doob-Meyer decomposition for right-continuous supermartingales
We complete here the proof of Proposition 3.1, based on a personal communication with Nicole El Karoui.
Proof of Proposition 3.1. Let us start by considering the following reflected BSDEs with lower obstacle X on where N is again a càdlàg martingale orthogonal to W , and K is a càdlàg nondecreasing and predictable process. Since the obstacle X is assumed to be càdlàg, the wellposedness of such an equation is guaranteed by [4, Theorem 3.1].
Let us now prove that we have Y t = X t , a.s., for any t ∈ [0, τ ]. Let us argue by contradiction and suppose that this equality does not hold. Without loss of generality, we can assume that Y 0 > X 0 (otherwise, we replace 0 by the first time when Y > X + ι for some ι > 0). Fix then some ε > 0 and consider the following stopping time Since Y is strictly above X before τ ε , we know that K is identically 0 on [[0, τ ε ]], which implies that Consider now the following BSDE on By standard a priori estimates (see for instance [4,Rem. 4.1]), we can find a constant C > 0 independent of ε > 0 s.t.
But remember that X is an E g -supermartingale, so that we must have y 0 ≤ X 0 . Hence, we have obtained Y 0 ≤ X 0 + Cε, which implies a contradiction by arbitrariness of ε > 0.
The uniqueness of the decomposition is then clear by identification of the local martingale part and the finite variation part of a semimartingale.

A.2 Down-crossing lemma of E g -supermartingale
We provide here a down-crossing lemma for E g -supermartingales (defined in Section 3 with g satisfying (4) and (5) for some p > 1), which is an extension of Chen and Peng [7, Thm 6] (see also Coquet et al. [10,Prop. 2.6]). For completeness, we will also provide a proof. As in the classical case, g ≡ 0, it ensures the existence of right-and left-limits for E g -supermartingales, see Lemma A.2 below.
For any m > 0, we denote by E ±m σ,τ the non-linear expectation operator associated to the generator (t, ω, y, z) −→ ±m|z| and stopping times (σ, τ ) ∈ T 2 . Let J := (τ n ) n∈N be a countable family of stopping times taking values in [0, T ], which are ordered, i.e. for any i, j ∈ N, one has τ i ≤ τ j , a.s., or τ i ≥ τ j , a.s. Let a < b, X be some process and J n ⊆ J be a finite subset (J n = {0 ≤ τ 1 ≤ · · · ≤ τ n ≤ T }). We denote by D b a (X, J n ) the number of down-crossing of the process (X τ k ) 1≤k≤n from b to a. We then define D b a (X, J) := sup D b a (X, J n ) : J n ⊆ J, and J n is a finite set .
Lemma A.1 (Down-crossing). Suppose that the generator g satisfies (4) with Lipschitz constant L in y and µ in z, and (5) with p > 1. Let X ∈ X p be a E g -supermartingale, J := (τ n ) n∈N be a countable family of stopping times taking values in [0, T ], which are ordered. Then, for all a < b, Proof. First, without loss of generality, we can always suppose that τ 0 ≡ 0 and τ 1 ≡ T belong to J, and also that b > a = 0. Indeed, whenever b > a = 0, we can consider the barrier constants (0, b − a), and the Eḡ-supermartingale X − a, with generatorḡ t (y, z) := g t (y + a, z), which reduces the problem to the case b > a = 0. Now, suppose that J n = {τ 0 , τ 1 , · · · , τ n } with 0 = τ 0 < τ 1 < · · · < τ n = T . We consider the following BSDE By the comparison principle for BSDEs (see [30,Prop. 4]), and since X is an E g -supermartingale, it is clear that Solving the above linear BSDE (20), it follows that  Denote then by D u l (Y, J) (resp. D u l (Y , J)) the number of down-crossing of the process Y (resp. Y ) from the upper boundary u to lower boundary l. It is clear that D u l (Y, J) = D u l (Y , J). Notice that l t is decreasing in t, so that we can apply the classical down-crossing theorem for supermartingales (see e.g. Doob [14, p.446]) to Y , and obtain that Notice that |λ s | ≤ L, |η s | ≤ µ and (X T ∧ b) = (X T ∧ b) + − (X T ∧ b) − . Therefore, we have proved (19) for the case b > a = 0, from which we conclude the proof, by our earlier discussion.