\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\varvec{L}}}^{{\varvec{p}}}$$\end{document}Lp-Solutions and Comparison Results for Lévy-Driven Backward Stochastic Differential Equations in a Monotonic, General Growth Setting

We present a unified approach to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document}Lp-solutions (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p > 1$$\end{document}p>1) of multidimensional backward stochastic differential equations (BSDEs) driven by Lévy processes and more general filtrations. New existence, uniqueness and comparison results are obtained. The generator functions obey a time-dependent extended monotonicity (Osgood) condition in the y-variable and have general growth in y. Within this setting, the results generalize those of Royer, Yin and Mao, Yao, Kruse and Popier, and Geiss and Steinicke.


Introduction
The existence and uniqueness of solutions to a backward stochastic differential equation (BSDE) have been extensively investigated in many, but also various specifically chosen settings, partly due to certain applications in practice and partly also for theoretically interesting reasons. In this paper, we both unify and simplify the approach for a general BSDE framework driven by a Lévy process with a straightforward extension to more general filtrations. We show new comparison results and relax the assumptions known so far for guaranteeing unique L p -solutions, p > 1, to a BSDE with terminal condition ξ and generator f that satisfies a monotonicity condition. An L p -solution is a triplet of processes (Y , Z , U ) from suitable L p -spaces (defined in Sect. 2) which satisfies a.s.
for each t ∈ [0, T ], where W is a Brownian motion andÑ is a compensated Poisson random measure independent of W . The BSDE (1) itself will be denoted by (ξ, f ).

Related Works
For nonlinear BSDEs (ξ , f ) driven by Brownian motion, existence and uniqueness results were first systematically studied by Pardoux and Peng [21] with (ω, y, z) → f (ω, y, z) Lipschitz in (z, y) and ξ square integrable. The importance of BSDEs in mathematical finance and stochastic optimal control was further elaborated by various works, e.g., by El Karoui et al. [7] which consider Lipschitz generators, L psolutions and Malliavin derivatives of BSDEs in the Brownian setting. The ambition to weaken the assumptions on f and ξ to still guarantee a unique solution gave birth to a large number of contributions, where-in the case of a generator with Lipschitz dependence on the z-variable-at least a few should be mentioned herein: Pardoux [20] and Briand and Carmona [2] considered monotonic generators w.r.t. y with different growth conditions. Mao [15] used the Bihari-LaSalle inequality to generalize the growth condition. Briand et al. [3] proved existence and uniqueness of a solution in the case where the generator may have a general growth in the y-variable and both T 0 | f (s, 0, 0)|ds and ξ belong to L p for some p ≥ 1. Generalizing the driving randomness, Tang and Li [28] and many other papers studied BSDEs including jumps by a Poisson random measure independent of the Brownian motion. Treating BSDEs in the case of quadratic growth in the z-variable, a considerable amount of articles was published in the recent years starting from the seminal paper of Kobylanski [12] in 2000 to recent papers using BMO methods such as [4] in the Brownian case or also comparison theorems like in [9] who consider an additional Poisson random measure as driving noise. We skip detailed comments in the direction of quadratic growth BSDEs as we will not consider this setting in our article.
Recent and most relevant for the present paper are the results by Kruse and Popier [13] considering L p -solutions for BSDEs driven by Brownian motion, a Poisson random measure and an additional martingale term under a monotonicity condition. They included the case of random time horizons. Yao [29] studied L p -solutions to BSDEs with a finite activity Lévy process for 1 < p < 2 and used a generalization for the monotonicity assumption similar to the one of [15] and also used in Sow [26]. Generalizing the L p -assumptions for the monotonic generator setting, in [8] the existence (and uniqueness in [5]) of a solution was proved for a scalar linearly growing BSDE when the terminal value ξ admitted integrability of |ξ | exp μ 2 log(1 + |ξ |) for a parameter μ > μ 0 , for some critical value μ 0 > 0. Moreover, a counterexample in [8] shows that for the case μ < μ 0 the preceding integrability is not sufficient to guarantee existence. In the critical case μ = μ 0 , they prove existence and uniqueness of a solution assuming a uniform Lipschitz generator.

Main Contribution
Within our approach, the results shed new light on the extensive literature of BSDE existence and uniqueness results as follows. In [14], Kruse and Popier designed function spaces such that their results of [13] extend to 1 < p < 2. In the present article, we show that the BSDEs' solutions for 1 < p < 2 are even contained in the usual L p spaces as defined for p ≥ 2. Moreover, an additional martingale term M orthogonal to W andÑ as used by Kruse and Popier [13] can also be added to our setting as an extension of (1), with unknown variables (Y , Z , U , M), as the careful analysis in their paper shows how the bracket process [M] has to be treated in an a priori estimate. All the results we obtain are still valid in this extended setting-see Remark 1. Nonetheless, we decided to omit the presentation of the straightforward martingale term, since the main difficulty lies in the treatment of the compensated Poisson random measure. The paper of Geiss and Steinicke [10], placed in a one-dimensional L 2 -setting only, requires a linear growth condition on the generator and needs approximation results for the comparison theorem, while the present setting allows first of all general growth, but even uses a simpler approximation technique for the comparison theorem avoiding deep-lying measurability results and, for p ≥ 2, only requires comparison of the generators on the solution processes.
Furthermore, in contrast to [3], [13] and others, this article establishes the more general monotonicity (Osgood) condition with a nondecreasing, concave function ρ to relax the generator's dependence on y (see also Mao [15]). This includes, e.g., continuities of the type as the function y → −y log(|y|) possesses at y = 0. Using the general approach, similar a priori estimates are shown to still hold true in order to guarantee uniqueness of an L p -solution, p ≥ 2.
In addition, the results of Yao [29] are extended in the sense that we do not require the jump process to have a finite Lévy measure.
Hence, we close several gaps in the theoretical understanding of solutions to BSDEs driven by a Lévy process, for the class of generators which are Lipschitz in the z-and uvariables. The more delicate techniques needed for this paper's approach for existence and uniqueness are inspired by the ideas of [3] along with [13] and [7]. In that spirit, before starting the main proofs, we obtain useful a priori estimates for the solution processes. For the comparison theorem, we enhance ideas and simplify proofs from [10] and [23].

Structure of the Paper
This paper is organized in the following way: First we establish the setting in Sect. 2 and state the assumptions and the main theorem (Sect. 3). After developing a priori estimates in Sect. 4, we finally prove existence and uniqueness of L p -solutions for p > 1 in Sect. 5 and end up with the comparison results for p ≥ 2 and 1 < p < 2 in Sect. 6.

Setting
Throughout the paper, we will use the following setting: In dimension d ≥ 1, let | · | denote the Euclidean distance. For x, y ∈ R d , we write x, y = d i=1 x i y i , and for z ∈ R d×k , k ≥ 1, we denote |z| 2 = trace(zz * ). The operations min(a, b) and max(a, b) will be denoted by a ∧ b and a ∨ b.
Let X = (X t ) t∈[0,T ] be a càdlàg Lévy process with values in R d on a complete probability space (Ω, F , P) with Lévy measure ν. By (F t ) t∈[0,T ] , we will denote the augmented natural filtration of X and assume that F = F T . Equations or inequalities for objects on these spaces are considered up to P-null sets. Conditional expectations E [ · |F t ] will be denoted by E t .
The Lévy-Itô decomposition of X can be written as where a ∈ R d , Σ ∈ R d×k with full column rank, W is a k-dimensional standard Brownian motion and N (Ñ ) is the (compensated) Poisson random measure corresponding to X . For the general theory of Lévy processes, we refer to [1] or [24]. This setting can be adapted to a pure jump process, if one sets Σ = 0 and omits the stochastic integrals with respect to W in the BSDE. Generalizing the above setting slightly, F can be assumed to be generated by a k-dimensional Brownian motion W and an independent (from W ) compensated Poisson random measureÑ on R \{0} for some ≥ 1, and the assumption of a driving process X can in principle be omitted. For convenience, however, we will stick to the setting emerging from a driving Lévy process X .
-Let S p denote the space of all (F t ) t∈[0,T ] -progressively measurable and càdlàg processes -We define L p (W ) as the space of all progressively measurable processes Z : -L loc (W ) denotes the space of R d×k -valued progressively measurable processes, such that for every t > 0, t 0 |Z s | 2 ds < ∞, P-a.s.
-With a slight abuse of notation, we define For F ∈ L p (Ω; L 1 ([0, T ])), we define The notions I F and K F are designed to make use of the simple properties, F = I F K F and I s. These properties are used, e.g., equation (15).
-We consider the terminal condition ξ to be an F T -measurable random variable and the generator to be a random function f : Definition 1 An L loc -solution to a BSDE (ξ, f ) with terminal condition ξ and generator f is a triplet Remark 1 An extension of our setting in the way of [14] is the following: Redefine the above spaces using a filtration (F t ) t∈[0,T ] on (Ω, F , P), which is assumed to be quasi-left continuous, satisfies the usual conditions and supports a kdimensional Brownian motion W and a compensated Poisson random measureÑ on R \{0}. Furthermore, we introduce the space M loc of càdlàg local martingales orthogonal to W andÑ and the space M of true martingale processes in M loc . Moreover, define and in the sense of Definition 1 and 2, let an L loc -solution (respectively, L p -solution) to a BSDE (2) be a tuple (1). As mentioned in the introduction, our main results in the following sections can also be shown within the extended setting producing some extra lines of technical computations.

Lévy Process with Finite Measure
The driving Lévy process, given by its Lévy-Itô decomposition (3), will be approximated for n ≥ 1 by The process X n has a finite Lévy measure. Note furthermore that the compensated Poisson random measure associated with X n can be expressed asÑ n = χ {1/n≤|x|}Ñ , where χ A denotes the indicator function of a set A. Let where N stands for the null sets of F . Denote by E n the conditional expectation E · |F n .

Main Theorem
With this setting in mind, we now state the main theorem based on the following assumptions, with a slight distinction for p ≥ 2 and p < 2, which turns out to be quite natural for the proofs. Instead of a Lipschitz condition, we require the weaker conditions (A3 ≥2 ) and, respectively (A3 <2 ), referred to as one-sided Lipschitz or monotonicity condition for the generator f .
(ii) The ρ-function appearing in the right-hand sides of (A3 ≥2 ) and (A3 <2 ) admits the following inequalities, which play important roles in the proofs: Proof For (ii), we see that, if |y| < 1, then |y| p−2 < 1 and by the concavity of ρ, For |y| ≥ 1, we have by the concavity of ρ, The case 0 < p ≤ 2 is similar.
Remark 3 (i) In (A3 <2 ), if β 2 is deterministic, we could impose the weaker condition q = 2 as described later in Remark 5. (ii) The following example is constructed in order to demonstrate the possibilities in this setting for d = 1, p > 1. All the involved expressions are chosen to exploit the assumptions on the coefficients which may be time-dependent, unbounded and some even random. The generator's dependence on y is not Lipschitz (not even one-sided Lipschitz) and of super-linear growth: where μ is given by μ(ω, t) = ∞ n=1 1 n 2 √ t−t n (ω) , with (t n (ω)) n≥1 being a numeration of the jumps of the trajectory t → X t (ω) of the Lévy process and μ(t, ω) = 0 if t → X t (ω) has no jumps, , when defined, 0 e l s e ,
We will prove this theorem in Sect. 5 after presenting necessary a priori estimates in the next section.

A Priori Estimates and Stability
Throughout the next sections, recall that f 0 (t) = f (t, 0, 0, 0), and that I | f 0 | and K | f 0 | are defined as in (4).

Remark 4
For the results in this section, it suffices to require a weaker condition than (A3 ≥2 ). We define this adapted assumption (a3 <2 ) with the same requirements, except one: We replace the monotonicity condition (5) by and analogously, we adapt the assumption (A3 <2 ) to derive the weaker (a3 <2 ) by replacing inequality (6), Lemma 1 If a sequence of random variables (V n ) n∈N in L p satisfies lim n→∞ E|V n | p = 0, then for a function ρ as in the assumptions, we have Proof This follows from the continuity of ρ, ρ(0) = 0 and the uniform integrability of (|V n | p ) n≥1 , since ρ(x) ≤ a + bx for some a, b > 0 and the above inequality shows that also (ρ(|V n | 2 ) p 2 ) n≥1 is a uniformly integrable sequence.
The following two propositions show that the norms of the Z and U processes can be controlled by expressions in Y and f 0 . Note that the bounds in Proposition 1 and Proposition 2 differ slightly, so that the application of Proposition 1 in Sect. 5 needs the assertion of Lemma 1. More precisely, there is a constant C > 0 depending on p, T , α, μ, β such that for Proof This proof generalizes the arguments in [3, Lemma 3.1].
Step 1: For t ∈ [0, T ] and n ≥ 1, define the stopping times Itô's formula implies from which we infer by (a3 ≥2 ) that Taking the power p 2 , we find a constant c 0 > 0 such that We continue our estimate (with another constant To estimate the above further, we have to split up the range of values of p ≥ 2.
We use the following inequality given, e.g., in [17, Theorem 3.2], which states that for a local martingale M, given by , there exists c 2 > 0 such that the following inequality holds for p ∈ ]0, 2]: Here, we will apply this inequality for p = p/2 to the martingale Note that we can estimate the square of the above integrand P ⊗ λ ⊗ ν-a.e. by We take suprema and expectations to get a constant c 3 > 0 such that Young's inequality (see Theorem 4 in Appendix) now gives us for an arbitrary Choosing now R such that 2c 3 Taking the limit for n → ∞ shows the assertion for 2 ≤ p ≤ 4. Case 2: p > 4 We start from (10) following the same lines of the previous case. In this case, the only difference is: [17,Theorem 3.2] states that for a local martingale M, given by For p = p 2 , we apply this inequality to the local martingale The first summand of (12) can be treated as in case 1. We focus on the second term which equals We can bound the integrands (as explained in (11)) by and Hence, we find a constant c 5 > 0, such that (13) is smaller than Using Young's inequality for the conjugate couple ( p 2 , p p−2 ), we have for arbitrary From here, similar steps as in case 1 conclude the proof.
The assertion holds true even if q = 2 in (a3 <2 ) since we do not use a higher integrability condition in the proof.
Proof We proceed as in the proof before until (9) and then infer by (a3 <2 ) in Remark 4, Taking the power p 2 , we find a constant c 0 > 0 such that We estimate further with c 1 > 0 With Young's inequality for ( 2 p , 2 2− p ) and a new constant c 2 > 0, we get From here on, the proof can be concluded similar to case 1 of Proposition 1.
From the above proposition, we now know how to bound Z and U in terms of Y and f 0 . For the core of the existence proof later, we need to control the Y part of the solution triplet by a bound depending only on ξ and f , which we will show in the sequel.

Proof
Step 1: Let Ψ (y) := |y| p and η = (η t ) 0≤t≤T ∈ L ∞ (Ω; L 1 ([0, T ])) be a progressively measurable, continuous process, which we will determine later. Itô's formula (see also [13,Proposition 2] where D 2 Ψ denotes the Hessian matrix of Ψ , By the argument in [13, Proposition 2], we can use the estimates trace(D 2 Ψ (y)zz * ) ≥ p|y| p−2 |z| 2 and leading to Using c z = 1 2 p(1 − p) and c u = p(1 − p)3 1− p , (a3 ≥2 ), Remark 2(ii)(a), Young's inequality for arbitrary R z , R u > 0 with the conjugate couple (2, 2), Young's inequality once more for the expression (4)) and the couple ( p p−1 , p), we find We Now, we omit e t 0 η(s)ds |Y t | p and take expectations, Hence, we find a constant c 0 > 0, to end the step with Step 2: We take the same route as in the previous step until (15), with one difference: we keep the P(t) term, to get Now we set R z = p/(2c z ), R u = 1/2 and η = p(αρ(1) We can rewrite By Taylor expansion of | · | p (see [13,Proposition 2]) With the minus in front, we can omit the integral with respect to N (ds, dx) in (17), take suprema and end up with We proceed by estimating the expectation of these two suprema in the next step.
Step 3: For the first supremum, we apply the Burkholder-Davis-Gundy inequality ([11, Theorem 10.36]) giving c 2 > 0 and the first line of the following inequality. Then, we pull out sup s∈[t,T ] |Y s | p 2 from the ds-integral (and the squareroot) and finally use Young's inequality for arbitrary R > 0, to estimate In the last step, we used Young's inequality as above in (19) for some arbitrary R > 0 to get the constant c 5 > 0.
Step 4: With the last step's results, we continue from (18) We apply inequality (16) yielding We choose R = 2D, which implies that there is where we also used the concavity of ρ. Now, the Bihari-LaSalle inequality (see Theorem 5 in Appendix) finishes the proof.

Proof
Step 1: We begin this proof similarly to the case p ≥ 2: Let η be a progressively measurable process in L ∞ (Ω; L 1 ([0, T ])), which we will determine later. As carried out in detail in [13, Proposition 3], Itô's formula, applied to the smooth function u ε : x → (|x| 2 + ε) p 2 and taking the limit ε → 0 implies that for c 0 = p( p−1) The terms slightly differ from [13,Proposition 3]. The alternative expressions are due to the relation dÑ (ds, dx) = d N (ds, dx)−ν(dx)dt, used to split up the integrals w.r.t. those random measures accordingly in the limit procedure. We may do this as all relevant integrands appearing in the Itô formula for u ε and in the limit expression yield P-a.s. finite integrals. Their finiteness results from the convexity of u ε , the boundedness of its second derivative for all ε > 0 and from the fact that By the argument in [13, Proposition 3] we can use the estimate Using (a3 <2 ) and Young's inequality, we obtain for an arbitrary R z > 0, We choose R z = p c 0 and η = p μ + R z β 2 1 2 + ( p − 1)K | f 0 | , take expectations and omit the first term to arrive at yielding a constant C > 0 such that Step 2: In this step, we leave the argumentation lines of Kruse and Popier [13,14] and Briand et al [3], estimating several terms differently and using the integrability assumptions on β 2 . We start from estimating the suprema of the stochastic integrals appearing in (21) by similar means as in step 3 of the proof of Proposition 3, (18) - (20) which yields constants c, c 1 > 0, such that for an arbitrary R > 0 we get where again we used Young's inequality as well as that Taking suprema in (21) with the same choices R z = p c 0 and η = p μ + where we omitted the remaining positive integral that appear on the left-hand side after cancelling the integrals involving |Y s | p−2 |Z s | 2 and |Y (s)| p (not the one with ρ) on the right-hand side of (21) after substituting R z and η.
Using the above estimate (23) Now inequality (22) can be plugged in for the last parentheses to estimate, for another constant D > 0, We focus on the term E T t β 2 (s)|Y s | p−1 U s ds, which we estimate by for R 1 > 0 and a constant c 2 > 0 coming from Young's inequality for the couple ( p, p p−1 ). By the Cauchy-Schwarz inequality, we get Now we use the additional integrability of β 2 with a power q > 2. For the case β 2 ∈ L 2 ([0, T ]), see Remark 5 after the proof. Here, in the sequel we treat a nondeterministic β 2 where higher integrability is needed. It has the only purpose to obtain a factor containing T − t by Hölder's inequality. Indeed, we infer from (25) E sup

Now, by the boundedness of
T 0 β 2 (s) q ds, and applying Proposition 2, we get a constant D 1 > 0 such that Inserting (26) into inequality (24), we get for a new constantD > 0 E sup . Now, our goal for the next step is to divide [0, T ] into small parts in order to make the third term containing (T − t) small too.

Now, the Bihari-LaSalle inequality (Theorem 5) shows that there is a function h n such that
Performing the same steps as above for the interval [t n−2 , t n−1 ], we find a function h n−1 such that Iterating the procedure backwards in time, we end up with functions h 1 , . . . , h n , accumulating to a functionh, such that then follows from Proposition 2, concluding the proof.

Remark 5
In step 3, if β 2 is deterministic, we could impose the weaker condition, namely β 2 being only square-integrable (instead of in L q , for some q > 2). Then we do not need to apply Hölder's inequality to (25) in order to choose the division of pq q−2 is small. Instead we choose the partition such that the t i t i−1 β 2 (s) 2 ds become sufficiently small. With the technique from the two above a priori estimates in hand, we can now prove another key part for the existence proof: boundedness stability of the Y process, meaning that the solution process Y stays bounded, when the data (ξ , f ) have boundedness properties: p > 1 and (Y , Z , U ) be an L p -solution to the BSDE (ξ, f ).
Proof We copy the proofs of Proposition 3 and Proposition 4 for the cases 1 < p < 2 and 2 ≤ p, replacing the operator E by E [ · |F t ] considering the BSDEs on [t, T ], which leads to the estimates E sup s∈[t,T ] |Y s | p F t < C for all t ∈ [0, T ]. The assertion now follows from the monotonicity of the conditional expectation.

Proof of the Main Theorem 1
The proof basically follows the one in Briand et al. [3,Theorem 4.2]. For convenience of the reader, we give a detailed proof adapted to our more general setting. We consider only the case 1 < p < 2 as the case p ≥ 2 is similar but easier.
Step 1: Uniqueness Assume we have another solution (Y , Z , U ). Then Proposition 4 applied to the BSDE (0, g) with g(t, y, z, Step 2: In this step, we construct a first approximating sequence of generators for f and show several estimates for the solution processes. Assume that ξ, I | f 0 | ∈ L ∞ . As (A3 <2 ) is satisfied, the condition is also satisfied for the changed parameter μ = ρ(1)α + μ. We take the constant C appearing in Proposition 5 and choose an r > C.
Take a smooth real function θ r such that 0 ≤ θ r ≤ 1, θ r (y) = 1 for |y| ≤ r and θ r (y) = 0 for |y| ≥ r + 1 and define h n (t, y, z, u) := θ r (y) ( f (t, y, c n (z),c n (u) Here, c n ,c n are the projections x → nx/(|x| ∨ n) onto the closed unit balls of radius n, respectively, in R d×k and L 2 (ν).
Moreover, by property (A iv) and (7) we are able to apply Proposition 5 to get that Y n t ∞ ≤ r . Since Y n t is bounded by r , we get that (Y n , Z n , U n ) is also a solution to the BSDE (ξ, f n ), with f n (t, y, z, u) : = ( f (t, y, c n (z),c n (u) Comparing the solutions (Y n , Z n , U n ) and (Y m , Z m , U m ) for m ≥ n, we use the standard methods from (14)- (15), for the differences In this procedure, we replace the use of the monotonicity condition (A3 ≥2 ) in Proposition 4 by such that the same steps of the proof of Proposition 4 can be conducted to get a function h with (in the case for (A3 ≥2 ), we use the steps from Proposition 3 and Proposition 1). So ΔY tends to zero if does, which we will show next (in the case of (A3 ≥2 ), this follows from Proposition 3 with (8)).
Since |Y m t |, |Y n t | ≤ r , we estimate Because of the definition of f m , f n and since m ≥ n, the integrand is zero if |Z s | ≤ n, U s ≤ n and ψ r +1 (s) ≤ n and bounded by otherwise.
To show convergence of the integral of (28), we use the uniform integrability of the families Φ(|Z n | + U n ) n≥1 with respect to the measure P ⊗ λ, which follows from since by Proposition 4 and (A iv), there is r > 0 such that Z n 2 L 2 (W ) + U n 2 L 2 (Ñ ) < r . Therefore, as (28) (as sequence in n) is uniformly integrable with respect to P ⊗ λ, dominating the sequence f m (s, Y n s , Z n s , U n s ) − f n (s, Y n s , Z n s , U n s ) n≥0 , which approaches zero pointwisely, also (27) tends to zero as m > n → ∞. Hence, also ΔY tend to zero, showing that the (Y n , Z n , U n ) form a Cauchy sequence in S p × L p (W ) × L p (Ñ ) and converge to an element (Y , Z , U ).
Step 3: We now show that (Y , Z , U ) satisfies the BSDE (ξ, f ), for ξ, I | f 0 | ∈ L ∞ , as supposed in step 2. The stochastic integral terms of the BSDEs (ξ, f n ) with solution (Y n , Z n , U n ) converge to the corresponding terms of the BSDE (ξ, f ) also in probability. It is left to show that, at least for a subsequence, For an appropriate subsequence, all other terms of the BSDEs converge almost surely. W.l.o.g, this subsequence is assumed to be the original one. Hence, we know that there is a random variable V t such that then for all t ∈ [0, T ], we have P-a.s, The same assertion follows from an equivalent formulation for f , requiring (29) being satisfied for f .

Proof
The basic idea for this proof was inspired by the one of Theorem 8.3 in [6] and is an extension and simplification of the one in [10].
Step 1: We use the conditional expectation E n (see Sect. 2.2) on the BSDEs (ξ, f ) and (ξ , f ) to get (for the BSDE (ξ, f )) progressively measurable) processes that equal the conditional expectation P-a.s. for almost all t ∈ [0, T ] (or λ ⊗ ν almost every (t, x) ∈ [0, T ] × R 0 for the U -process). In the case of Y , (E n Y t ) t∈[0,T ] denotes a progressively measurable version of this process. For bounded or nonnegative processes, the construction of such processes can be achieved by using optional projections with parameters (see [18] for optional projections with parameters and [10] for the mentioned construction). In the present case, we are confronted with merely integrable processes: Y , Z , R 0 |U · (x)| 2 ν(dx) are integrable and hence also f (s, Y s , Z s , U s ) ∈ L 1 (W ). The construction of a progressively measurable version for the processes at hand can be found in Lemma 2 and Remark 7 in Appendix.
Moreover, assume for the rest of the proof that the coefficient μ of f is zero: If this was not the case, we could use the transformed variables Step 2: We use Tanaka-Meyer's formula (cf. [22,Chapter 4, Theorem 70 and Corollary 1]) for the squared positive part function (·) 2 + to see that for η := 18β 2 , Here, M(t) is a stochastic integral term with zero expectation which follows from Y , Y ∈ S 2 . Moreover, we used that on the set We denote further differences by Δ n ξ := E n ξ − E n ξ , Δ n Z := E n Z − E n Z and Δ n U := E n U − E n U . Observing that the right-hand side increases if we only consider Δ n Y s > 0 in the ds-integral, we take means to come to We split up the set {1/n ≤ |x|} into Taking into account that ξ ≤ ξ ⇒ E n ξ ≤ E n ξ , we estimate We focus on (Δ n Y s ) Since (A3 ≥2 ) implies the Lipschitz property in the u-and z-variables, we infer, inserting and subtracting the same terms, We estimate, inserting and subtracting terms again, then using (29), Next we apply Jensen's inequality in two dimensions for the product of positive random variables and also (A3 ≥2 ) and Young's inequality to arrive at Taking together inequalities (31), (32), (33) and (34), we get with Young's inequality again that Therefore, (30) evolves to We cancel out terms and end this step with the estimate Step 3: We assume without loss of generality that the integrals converge to zero. For domination, we use where we applied that T 0 η(s)ds < C a.s., Doob's martingale inequality and that there is b > 0 such that for x ≥ 0 : ρ(x) ≤ 1 + bx.
For m ≥ 0 with δ ρ − 1 m > 0, let us now choose N m ∈ N large enough, such that for n ≥ N m : For such an n, we get that In the same way, one can choose m, N m ∈ N also large enough such that for all n ≥ N m , Similarly, by martingale convergence E n Z s → Z s and a domination argument, we can conclude that for n ≥ N m (N m may have to be rechosen large enough), since the left-hand sides tend to zero, while the right-hand sides converge to δ y . The same estimates hold for Z and U as well. Hence, applying (36) and (37)  The term e T 0 η(τ )dτ is P-a.s. bounded by a constant C > 0. Thus, by the concavity of ρ n := ρ + ν({1/n ≤ |x|})id, which satisfies the same assumptions as ρ, we arrive at Then, the Bihari-LaSalle inequality (Theorem 5) shows that E(Δ n Y t ) 2 + = 0 for all t ∈ [0, T ].
Step 5: Steps 1-4 granted that E n Y t ≤ E n Y t for n greater than a certain value, for all t ∈ [0, T ]. By martingale convergence, E n Y t converges almost surely to the solution Y t of (ξ, f ) at time t and E n Y t converges to the solution Y t of (ξ , f ). Hence, in the limit we have Y t ≤ Y t , and the theorem is proved.
In Theorem 3, we state a version of the above theorem for the case 1 < p < 2. The difference to Theorem 2 is that here we cannot compare the generators on the solution only. If one wants to keep the comparison of the generators on the solution, but accepts a slightly stronger condition than (29), given as (H comp )in [13], for a predictable process γ = γ y,z,u,u , such that −1 ≤ γ t (x) and |γ t (u)| ≤ ϑ(u), where ϑ ∈ L 2 (ν), then the proof of [13,Proposition 4] can also be conducted for generators satisfying the conditions (A3 ≥2 ) or (A3 <2 ). Then, for all t, Y n,t → Y t , Y n,t → Y t , P-a.s., and Y t ≤ Y t , P-a.s. follows. (V , V , μ) be a σ -finite measure space and K : Ω × [0, T ] × V → R be a P ⊗ V -measurable process such that for λ ⊗ μ-almost all (t, x) ∈ [0, T ] × V ,
Proof For m ≥ 1, let K m := m |K |∨m K be a bounded approximation for K . For each m, we can construct the process K m n , as described above the lemma. We intend to let m → ∞. By the definition of the optional and predictable projections, we have that for all (t, x) ∈ [0, T ] × V , K m n (t, x) = E t− E n K m (t, x), P-a.s., where we emphasize that E t− is the conditional expectation with respect to the σalgebra F t− = σ 0≤s<t F s and E n denotes the expectation with respect to the σ -algebra F n . Using the assumption E|K (t, x)| < ∞ for λ ⊗ μ-a.a. (t, x), we find by dominated convergence that for a set Since (V , V , μ) is σ -finite, it follows that the sequence (K m n ) m≥1 converges also locally in the measure P ⊗ λ ⊗ μ, from which we then infer that we can extract a subsequence (K m k n ) k≥1 that converges P ⊗ λ ⊗ μ-a.e. Without loss of generality, we assume that (K m n ) m≥1 itself converges P ⊗ λ ⊗ μ-a.e. This means that one can find an F ⊗ B([0, T ]) ⊗ V -measurable processK n and a set M ∈ F ⊗ B([0, T ]) ⊗ V with P ⊗ λ ⊗ μ ((Ω × [0, T ] × V )\M) = 0 such that for all (ω, t, x) ∈ M, lim m→∞ K m n (ω, t, x) =K n (ω, t, x). On the σ -algebra we have that the processK n restricted to the set M is the limit of the processes (K m n ) restricted to M and is therefore (P ⊗ V ) M -measurable. In the sense of [25, Theorem 1], we can then extend the restriction of the processK n to a P ⊗ V -measurable process K n .
To show that the process K n satisfies condition (41) for λ ⊗ μ-a.a. (t, x) ∈ [0, T ] × V , one can use the dominated convergence theorem for conditional expectations,