Backward stochastic Volterra integral equations with jumps in a general filtration

In this paper, we study backward stochastic Volterra integral equations introduced in [26, 45] and extend the existence, uniqueness or comparison results for general filtration as in [31] (not only Brownian-Poisson setting). We also consider Lp-data and explore the time regularity of the solution in the It{\^o} setting, which is also new in this jump setting.


Introduction
The aim of this paper is to extend or to adapt some results concerning backward stochastic Volterra integral equations (BSVIEs in short). To the best of our knowledge, [26,45,46] were the first papers dealing with BSVIEs and the authors considered the following class of BSVIEs: W is a k-dimensional Brownian motion, f is called the generator or the driver of the BSVIE and Φ is the free term (or sometimes the terminal condition). Filtration F is the completed filtration generated by the Brownian motion. They proved existence and uniqueness of the solution (Y, Z) (M-solution in [46]) under the natural Lipschitz continuity regularity of f and square integrability condition for the data. Let us focus on two particular cases. If f and Φ are not dependent on t, we obtain a backward stochastic differential equation (BSDE for short): Since the seminal paper [32], it has been intensively studied (see among many others [11,16,34,39]). Expanding the paper [15], Papapantoleon et al. [31] studied BSDEs of the form: Here the underlying filtration F only satisfies the usual hypotheses (completeness and right-continuity). The exact definition of processes B, X • , π is given in Section 2. Roughly speaking, X • is a square-integrable martingale, π is an integer-valued random measure, such that each component of X • is absolutely continuous w.r.t. B and the disintegration property given B holds for the compensator ν of π . Martingale M naturally appears in the martingale representation since no additional assumption on filtration F is assumed. Their setting contains the particular case where X • = W and π is a Poisson random measure (Ex. 2.1), but also many others (see the introduction of [31]). The second particular case of (1.1) is called Type-I BSVIE: The extension to L p -solution (1 < p < 2) for the Type-I BSVIE (1.3) has been done in [41]. In the four papers [26,41,45,46], filtration F is generated by the Brownian motion W . In [44], the authors introduced the jump component π. In the filtration generated by W and the Poisson random measure π, they consider: and prove the existence and uniqueness of the solution in the L 2 -setting. The result has been extended in [17,30,37] (see also [28] for the Lévy case). BSVIEs were also studied in the Hilbert case [3], with additional perturbation in the Brownian setting [13,14], in the quadratic case [40], as a probabilistic representation (nonlinear Feynman-Kac formula) for PDEs [43]. Their use for optimal control problem has been well known since the seminal paper [45]; see for example the recent paper [27]. Let us also mention the survey [47].
Combining all these papers, here we want to deal with a BSVIE of the following type 1 : Filtration F and processes B, X • and π satisfy the same conditions as in [31]. The unknown processes are the quadruplet (Y, Z, U, M ) valued in R d+(d×k)+d+d such that Y (·) is F-adapted, and for (almost) all t ∈ [0, T ], (Z(t, ·), U (t, ·)) are such that the stochastic integrals are well-defined and M (t, ·) is a martingale. This BSVIE is called of Type-II. We also consider the Type-I BSVIE: Let us point out that in [31], the authors consider two different types of BSDEs: equation (1.2) and the following one: The only difference concerns the dependence of f w.r.t. Y . Since B is assumed to be random and càdlàg 2 , both cases are not equivalent. However the method of resolution in the second case is not adapted for BSVIEs. This case is left for further research.

Main contributions
Let us outline the main contributions of our paper compared to the existing literature. First we prove the existence and uniqueness of the adapted solution of the Type-I BSVIE (1.5) in the L 2 -setting (Thm. 3.3 and Prop. 4.3). This first result generalizes the prior results (of course only some of them) since we only assume that the filtration is complete and right-continuous. This is the reason fer the presence of the càdlàg process B and of the additional martingale term M in (1.5).
As explained in the introduction of [46], for Type-II BSVIEs, the notion of M-solution (see Def. 3.6 below) is crucial to ensure the uniqueness of the solution. To define the terms Z and U on the set ∆(R, T ) = {(t, s) ∈ [R, T ] 2 , R ≤ s ≤ t ≤ T }, the martingale representation is used. Since B is random, it is not possible to control the terms Z and U obtained by the martingale representation in a tractable way. We detail this point in Section 3.2.1. However if B is deterministic, we prove existence and uniqueness of the M-solution of the Type-II BSVIE (1.4) in the L 2 -setting (Thm. 3.9).
Our proofs are based on a fixed point argument in the suitable space. Thereby we impose some particular integrability conditions on the free term Φ and on process f (t, 0, 0, 0).
In the BSDE theory, many papers deal with L p -solution (instead of the square integrability condition on the data); see in particular [8,23,24,34] which deal with L p -solution for BSDE. Insofar as we know, such an extension does not exist for the general BSDE (1.2). This is the reason why we consider the Itô setting where B t = t in Section 2.2. This denomination comes from [1,2]. Thus X • is a Brownian motion W and π = π is a Poisson random measure with intensity measure µ. The Type-I BSVIE (1. For this BSVIE, we provide existence and uniqueness of an adapted solutions in L p -space of (1.6) (Thm. 3.10).
To the best of our knowledge, there is no existence and uniqueness result for BSVIEs with L p coefficients in a general filtration. Another contribution is the study of the regularity of the map t → Y (t). For the solution of BSDE (1.2), from the càdlàg regularity of all martingales, Y inherits the same time regularity. For BSVIE, we only require that the paths of Y are in L 2 (0, T ) (or in L p (0, T )). Essentially because we assume that Φ and t → f (t, . . .) are also only in L 2 (0, T ). In [46], it is proved that under weak regularity conditions for the data, then the solution of (1.1) t → Y (t) is continuous from [0, T ] to L 2 (Ω). Let us stress that Malliavin calculus is used to control the Z(s, t) term in the generator. Similarly we show that the paths t ∈ [0, T ] → Y (t) ∈ L p (Ω) of the solution of BSVIE (1.6) are càdlàg if roughly speaking Φ and t → f (t, . . .) satisfy the same property. However this first property does not give a.s. continuity of the paths of Y in general. Getting an almost sure continuity is a more challenging issue and is proved in [44] for BSVIE (1.1) when f does not depend on Z(s, t), assuming a Hölder continuity property of t → f (t, . . .) for a constant Φ(t) = ξ. To understand the difficulty, let us evoke that if f does not depend on y, the solution Y of BSVIE (1.6) is obtained by the formula: Y (t) = λ(t, t) where λ(t, ·) is the solution of the related BSDE parametrized by t. In the Brownian setting, a.s. s → λ(t, s) is continuous. Using the Kolmogorov continuity criterion, the authors show that (t, s) → λ(t, s) is bi-continuous, which leads to a continuous version of Y . Insofar as we know, there is not an equivalent result to the Kolmogorov criterion for càdlàg paths. Hence we assume that the free term Φ and the generator f are Hölder continuous. Thus we sketch the arguments of [44] to obtain that a.s. the paths of Y are càdlàg (Thm. 3.11) if we know that the data Φ and f are Hölder continuous w.r.t. t, meaning that the jumps only come from the martingale parts in the BSVIE. Relaxing the regularity of the data is still an open question.
In the Itô setting, BSVIE (1.4) becomes: It would be natural to prove existence and uniqueness of a solution in L p , as Theorem 3.10 for BSVIE (1.6). Nonetheless there is an issue again. Since Z is integrated w.r.t. the Brownian motion, the natural norm on Z is But this norm is not symmetric w.r.t. (t, s), except for p = 2. Despite our efforts, we still cannot overcome this problem. Moreover for p < 2, it is well known that Burkholder-Davis-Gundy inequality does not apply without continuity. Therefore the extension to p = 2 seems difficult to prove and is left for further research. We only give the result for p = 2 (Prop. 3.12), which is a corollary of Theorem 3.9. However the proof can be done following the outline of [46]. A real issue for BSVIEs concerns the comparison principle. In the BSDE theory, the comparison principle holds under quite general conditions (see e.g. [23,24,31,34]). Roughly speaking, the comparison result is proved by a linearization procedure and by an explicit form for the solution of a linear BSDE. However in the setting of [31], the comparison is more delicate to handle because of the jumps of the process B. For BSVIEs, these arguments fail and comparison is a difficult question. Paper [42] is the most relevant paper on this topic. It provides comparison results and gives several counter-examples where comparison principle fails. Of course all their counter-examples are still valid in our case; thereby we do not have intrinsically better results. On this topic our contribution is to extend the comparison results for the Type-I BSVIE (1.5) (Props. 6.3 and 6.4). Somehow we show that the additional martingale terms do not impair the comparison result. Note that we have to take into account that the driver f (s, Y (s), Z(s), U (s)) is a priori optional (compared to f (s, Y (s−), Z(s), U (s))). Thus the linearization procedure should be handled carefully.
Finally we prove a duality principle for BSVIE (1.7) provided we know that the solution X of the forward SVIE is itself càdlàg (see [35]). Note the importance of the time regularity here. This result is the first step for comparison principle for this kind of BSVIEs.

Breakdown of the paper
The paper is broken down as follows. In the first section, we give the mathematical setting and set out some results concerning the existence and uniqueness of the solution of BSDE (1.2). We also explain what we mean by Itô's setting. In the second part, we present our assumptions in details and the existence and uniqueness results concerning BSVIEs (1.4), (1.5), (1.6) and (1.7).
The proofs of these next two sections are presented in the next two sections. In Section 4, we consider the general case and the proofs are essentially based on a fixed point argument as in [31,45]. The Itô setting is developed in Section 5 (L p -solution (p > 1) and time regularity); for existence and uniqueness, the arguments are close to those of [46].
The last section is devoted to the comparison results for Type-I BSVIEs with their proofs, together with the duality result for Type-II BSVIE in Itô's framework.
Finally we set out some proofs and auxiliary results in the appendix.

Remaining open questions
Some open problems are addressed here. First of all, the existence and uniqueness of an M-solution for BSVIE (1.4) is not proved yet, except for a deterministic characteristic B. The second question concerns the comparison principle, at least for Type-I BSVIE (1.5), when B is not continuous. Finally the L p -theory for BSDE (1.2) and thus for BSVIEs (1.4) or (1.5) is a natural question.

Setting, notations and BSDEs
On R d , |.| denotes the Euclidean norm and R d×k is identified with the space of real matrices with d rows and k columns. If z ∈ R d×k , we have |z| 2 = Trace(zz * ). For any metric space G, B(G) is the Borel σ-field.
Our probabilistic setting is the same as that of Papapantoleon et al. [31]. The main notations are set out but the details can be found in this paper, especially in Section 2, and are left to the reader. Throughout this paper, we consider a filtered probability space (Ω, F, P, F = (F t ) t≥0 ) such that it is a complete stochastic basis in the sense of Jacod and Shiryaev [19]. Without loss of generality, all semimartingales are supposed to be càdlàg, that is they have a.s. right continuous paths with left limits. Space H 2 (R p ) denotes the set of R p -valued, square-integrable F-martingales and H 2,d (R p ) is the subspace of H 2 (R d ) consisting of purely discontinuous square-integrable martingales. P is the predictable σ-field on Ω × [0, T ] and P = P ⊗ B(R n ). On Ω = Ω × [0, T ] × R n , a function that is P-measurable, is called predictable.
The required notions on stochastic integrals are recalled in Section 2.2 of [31]. In particular for for B predictable non-decreasing and càdlàg, the stochastic integral of Z w.r.t X, denoted Z · X or 0 Z s dX s , is defined on space H 2 (X) of predictable processes Z : Ω × R + → R d×k such that Moreover π X is the F-optional integer-valued random measure on R + × R m defined by π X (ω; dt, dx) = s>0 1 ∆Xs(ω) =0 δ (s,∆Xs(ω)) ( dt, dx).
Stochastic integral U π X of U w.r.t. π X , compensator ν X of π X , compensated integer-valued random measure π X and stochastic integral U π X of U w.r.t. π X are also defined on space H 2 (X) of F-predictable processes Remember that all definitions are only summarized here; all details can be found in [19,31] for the interested reader.
In the rest of the paper, are fixed: Here X T is the process stopped at time T : X T t = X t∧T , t ≥ 0. M π · P is the conditional F-predictable projection on π (see [31], Def. 2.1). To simplify the notations, π = π (X ) T , π = π (X ) T . Keep in mind that ν is the compensator of π , that is the F-predictable random measure on R + × R m for which E[U µ ] = E[U ν ] for every non-negative F-predictable function U . -A non-decreasing predictable and càdlàg B such that each component of X • is absolutely continuous with respect to B and the disintegration property given B holds for the compensator ν , that is there exists a transition kernel K such that ν (ω; dt, dx) = K t (ω; dx) dB t (2.1) (see [31], Lem. 2.9). This property is called assumption (C) in [31]. -b is the F-predictable process defined in Remark 2.11 of [31] Process B is the first component of the characteristics of semimartingale X (see [19], Def. II.2.6).
Example 2.1. In Section 5, X • is the k-dimensional Brownian motion W , B t = t, b = 1 and π is the compensated Poisson random measure π, with the compensator ν( dt, dx) = dtµ( dx), where µ is σ-finite on R m such that In this particular case, K t (ω; dx) = µ( dx). These spaces are classically used for BSDEs with Poisson jumps (see among others [11,23]).
Example 2.2. The previous example can be generalized to the case where the compensator of π is random and equivalent to the measure dt ⊗ µ(ω, dx) with a bounded density for example (see the introduction of [4]).
Example 2.3. In Section 3.3 of [31], the authors cite the counterexample of [9]. They also provide two other examples just after their Remark 3.19.
The notion of orthogonal decomposition plays a central role here. Inspired by [19] and defined in Definition 2.2 of [31], if Y ∈ H 2 (R d ), then the decomposition The statements ( [31], Props. 2.5 and 2.6, Cor. 2.7) give the existence and the uniqueness of such orthogonal decomposition.

Setting and known results for BSDE (1.2)
In the rest of the paper, A is the non-increasing, F-predictable and càdlàg process defined by The process α : (Ω × R + , P) → R + may change through the paper. For some β ∈ R, let us describe the spaces used to obtain the solution of BSDE (1.2). For ease of notation, the dependence on A is suppressed.
We try to find the solution of BSDE (1.2) in the product space Let us introduce some additional notations. For an F-predictable function U , we specify where K is the kernel assigned by (2.1). Hence And The justification of the definition of the norms on the previous spaces is given by Lemma 2.12 of [31].
where for every (t, ω) ∈ R + × Ω As in Lemma 2.13 of [31], we designate Remark 2.5. In the setting of Example 2.1, U ≡ 0 and Then the generator can be defined on L 2 µ (R m ) instead of H in conditions (F2) or (H2). Now let us describe the conditions on parameters (ξ, f ) for BSDE (1.2).
(F4) For the same δ as in (F1), 0 denotes the null application from R m to R.
Let us evoke the existence and uniqueness result of [31].

The Itô setting and related BSDEs
In Sections 3.3 and 5, the processes are assumed to be Itô's semimartingales in the sense of Definition 1.16 of [1]. Hence the process B is now deterministic and equal to B t = t. SemimartingaleX can be represented by the Grigelionis form. 3 Up to some modifications in the generator 4 , BSDE (1.2) takes the next form: where -W is a Brownian motion -π is a Poisson random measure on [0, T ] × R m , with intensity dt ⊗ µ( dx).
Let us first evoke some standard notations.
For BSDE (1.2), the L p -theory has not been developed yet. But in the Itô case the next result is proved in [24]. Let us reinforce condition (H3): (H3*) There exists a constant K such that a.s. for any s ∈ [0, T ] and t ∈ [0, s], Proposition 2.7. Assume that for any (y, z, ψ), f (·, y, z, ψ) is progressively measurable and that (H2) and (H3*) hold. If 3 In general only on a very good filtered extension of the original probability space. But using the remarks of Section 1.4.3 in [1], we can assume that this form ofX holds on the original filtered probability space. 4 In general we should take into account a possible degeneracy of the coefficients. This possibility leads to a non Lipschitz continuous new version of the generator f . We ignore this trouble here.
there exists a unique solution (Y, Z, U, M ) in D p (0, T ) to BSDE (2.5). Moreover for some constant C = C p,K,T Condition (H3*) is certainly too strong for this result and could be relaxed. Thereby our results concerning BSVIEs in the Itô setting could be extended to non Lipschitz-continuous w.r.t. y driver f as in [44] (using Mao's condition with a concave function ρ) or if K becomes a function of (ω, s) with a suitable integrability condition (see [46], condition (3.13)). Nonetheless such extensions would increase the length of the paper and are left for further research.

Notations, definitions and statement of our main results for BSVIEs
Concerning BSVIEs, the notations of [46] are kept. They are only adapted to our setting and thus some details are skipped (see [46], Sect. 2.1 for interested readers).
This notation should not be confused with the jump of a process.
Recall that A is the non-negative càdlàg non-decreasing and measurable process defined by (2.2). Some β ∈ R + and some 0 ≤ δ ≤ β are fixed. Again to simplify the notations, the dependance on A is suppressed. First The above space is for the free term Φ(·) (for which the F-adaptiveness is not required). When F-adaptiveness is required, that is for Y (·): To control the martingale terms (Z, U, M ) and to define the notion of solutions, Type-II BSVIE (1.4) and Type-I BSVIE (1.5) are distinguished.

Existence and uniqueness for Type-I BSVIEs
To control the martingale terms (Z, U, M ) in the Type-I BSVIE (1.5), some other spaces are needed. The set of processes M (·, ·) such that for t ∈ [R, T ], M (t, ·) = {M (t, s), s ≥ t} belongs to H 2 (R d ) and In the particular case where Note that we strongly rely on the fact that we work on ∆ c (R, T ) and that A t is F t -measurable to obtain this equality. For Type-II BSVIEs this point is an issue. Similarly for U (t, ·) ∈ H 2,  In this case U ∈ H 2, δ≤β (∆ c (R, T )). To lighten the notations, the martingale for t ≤ u ≤ T is introduced, such that BSVIE (1.5) becomes Using Corollary 2.7 of [31], M belongs to H 2 δ≤β (∆ c (R, T )) if and only if the triplet (Z, U, M ) belongs to Last the product space is specified, with the naturally induced norm. If δ = δ(β) is a known function of β, S 2 δ≤β (∆ c (0, T )) is denoted by S 2 β (∆ c (0, T )).

Adapted solution for Type-I BSVIE
Let us adjust the notion of solution developed in [46] to our setting.
Our assumptions and the statement of our results depend on the next quantities. Let us define for δ < γ ≤ β the function The next technical lemma is equivalent to Lemma 3.4 in [31]. Its proof is set out in the Appendix.
is the unique solution on (0, β) of the equation: For M f (β) < 1/2, the next quantities are considered Let us precise the assumptions on the free term Φ and on the generator f of the BSVIE (1.5).
and for any fixed (t, y, z, u) the process f (t, ·, y, z, u) is progressively measurable. Moreover there exist To simplify the notation in the sequel: There exists f > 0 such that for any t ∆A r (ω) ≤ f, for dB ⊗ dP − a.e. (r, ω).
(H4) For the same β as in (H1), with the constant δ * (β) of the above Lemma 3.2 (H5) The following set is assumed to be equal to [0, T ].
Coming back to BSVIE (1.5), our main result on this BSVIE is now stated Let (Φ,h) be a couple of data each satisfying the above assumptions (H1) to (H5).
Then using the notation (3.1) the following stability result is verified: A comment on condition (3.5). The first part κ f (δ) < 1/2 comes from Theorem 2.6 (see [31], Thm. 3.5). The second condition is sufficient to have existence and uniqueness of the solution of the Type-I BSVIE if f does not depend on Y (see the BSVIE (4.4) and Prop. 4.3 below). The last part Σ f (β) < 1 ensures the existence and uniqueness thanks to a fixed point argument.

Remark 3.4 (Large values of β).
Note that for β large, κ f (δ) < 1/2 and M f (β) < 1/2 if and only if 18ef < 1 (as in [31]). And since For a random process B, the time integral and the expectation cannot be switched and a precise description of the set T Φ,f δ is not easy. But if B is deterministic, we deduce that dB-almost every t ∈ [0, T ] belongs to T Φ,f δ . In the Brownian-Poisson setting (Ex. 2.1), dB is the Lebesgue measure. If B is piecewise-constant as in the second example after Remark 3.19 of [31] with deterministic jump times, a similar dB-a.e. property is true. In other words (H5) is too strong in a lot of cases, but it makes the presentation of our results correct and easier in the general setting.

Extension for Type-II BSVIEs
Here the objective is to expand some results to Type-II BSVIEs (1.4). As explained in the introduction of [46], for Type-II BSVIEs, the notion of M-solution is crucial to ensure the uniqueness of the solution; uniqueness of an adapted solution fails. Roughly speaking, there is an additional freedom on ∆(0, T ). To avoid this problem, the next definition of M-solution is formulated in [46].
Note that the notion of M-solution of [46] is kept, where the letter M stands for "a martingale representation" for Y (t) to determine Z(·, ·), U (·, ·) and M (·, ·) on ∆[S, T ]. It should not be confused with the orthogonal martingale part M . As in [46], any M-solution on [S, T ] is also an M-solution on [S, T ] withS ∈ (S, T ).

The randomness of B is an issue
This notion of M-solution implies that to define the terms Z and U on the set ∆(R, T ) = {(t, s) ∈ [R, T ] 2 , R ≤ s ≤ t ≤ T }, the martingale representation is used. In [46] it takes the form: for almost every t ∈ [R, T ], . Integrating the relation (3.8) between R and t and using Fubini's theorem leads to In other words an M-solution (in the sense of [46]) provides a "integrated martingale"-solution.
For BSVIE (1.4) we only expect the Cauchy-Schwarz inequality yields to , and the martingale representation leads to The next estimate can easily be obtained: for any 0 < δ < β But this weak norm on M is not sufficient to control Z and U in the generator of BSVIE (1.4). Since process A is supposed to be only predictable, we cannot claim that It leads to a major issue. Moreover the cunning used in [31]: is useless here since e δAu is F u -measurable and not F S -measurable. Due to this reason, we cannot address an existence and uniqueness result for BSVIE (1.4) in this general setting, but only for deterministic processes A and B.

Type-II BSVIE for a deterministic process A
Therefore we restrict ourselves to the case where B is deterministic. Note that from Theorem II.4.15 of [19], semimartingale X has deterministic characteristics (that is B and the two other components) if and only if it is a process with independent increments. We also assume that α (see Hyp. (H3)) is deterministic.
For the free term Φ and the first component Y of the solution, spaces L 2 β,F T (0, T ) and L 2 β,F (0, T ) are conserved. But the definitions of Section 3.1 are adapted, since the martingale terms need to be controlled not only on ∆ c (0, T ), but on the whole set [0, T ] 2 . Thereby, space is defined as the set of processes M (·, ·) such that for dB-a.e. t ∈ [R, T ], M (t, ·) belongs to H 2 and In this case Z ∈ H 2,• δ≤γ (R, T ) (resp. U ∈ H 2, δ≤γ (R, T )). Finally we introduce the product space with the naturally induced norm. If δ = δ(γ) is a known function of γ, S 2 δ≤γ (R, T ) is denoted by S 2 γ (R, T ). Conditions (H2) and (H3) are modified as follows.
Remark 3.7. Let us emphasize that assumption (H2') implicitly implies that the process ν s is in fact F tmeasurable.
If we compare with the conditions imposed in [41,46], this assumption is stronger: Let us study (H1) and (H4) under the prior conditions. Hypothesis (H1) can be rewritten as follows: Moreover if A T < +∞, the estimate leads to (H1) and (H4). Let us again highlight that δ = δ * (β) is defined in Lemma 3.2.
. In other words A may be discontinuous, but with very small jumps.
To give an idea of the value of β, let us assume that f = 0, that is A is continuous. Then δ = δ * (β) = The condition (2.4) becomes And Some tedious computations show that β > 174 is sufficient for our condition (3.5). Nonetheless This leads to β > 1357 in order to satisfy (3.10).

In the Itô setting
In this framework (see Sect. 2.2) the Type-II BSVIE (1.4) becomes BSVIE (1.7): In the Brownian-Poisson L 2 -setting, BSVIE (1.6) was already studied in [44] and if there is only the Brownian component W , for p = 2, in [41]. These results are extended as follows: Theorem 3.10. For p > 1, assume that (H2) and (H3*) hold. Suppose that (3.11) Note that Let us emphasize that no regularity of the paths t → Y (t) is required; they are a priori neither continuous nor càdlàg. Component Y is only supposed to be in L p . If Y solves BSDEs (1.2) or (2.5), then it has the same regularity as the martingale part (if process B is continuous), thus a.s. it is a càdlàg process. For a BSVIE it is more delicate. In Theorem 4.2 of [46], the author shows that in the Brownian setting the BSVIE is continuous in L 2 (Ω), which does not mean that Y has a.s. continuous paths. Of course since t appears in generator f and in the free term Φ, some property on t → Φ(t) and t → f (t, . . .) has to be added. To obtain the time regularity for BSVIE (3.13), the author uses the Malliavin derivative to control the term Z(s, t) in the generator (see [46], Thm. 4.1 and 4.2). Hence to apply the same arguments, Malliavin calculus in the presence of jumps (see e.g. [7,12]) should be used. This point is left as future research and to avoid this technical machinery, let us study BSVIE (1.6). Integrability condition (3.11) is replaced by the stronger one: Finally instead of Φ ∈ L p F T (0, T ), we also assume that Under these two hypotheses and if f satisfies (H2) and (H3*), it is possible to deduce that Y is càdlàg from [0, T ] to L p F (Ω), providing some regularity assumption on t → Φ(t) and t → f (t, s, y, z, ψ) holds, as in Theorem 4.2 of [46] in the continuous setting (see Sect. A.3 in the appendix). Let us emphasize again that it does not mean that a.s. the paths of Y are càdlàg. Now the next regularity result is presented, which is the extension of Theorem 2.4 in [44]. The restriction p ≥ 2 is due to the dependence of the generator f on U . If it doesn't, the arguments of the proof lead to the same conclusion for p > 1. Following [24] it should be possible to extend the theorem for p > 1, but this point is left for further research.
Theorem 3.11. In addition to (H2) and (H3*), suppose that the generator satisfies for some p ≥ 2 and 0 < α < 1 such that αp > 1 and with > 0, uniformly in (ω, s, y, z, ψ): Then the solution Y of BSVIE (1.6) has a càdlàg version, still denoted by Y , such that Let us emphasize that our conditions imply that a.s. t → Φ(t) is continuous. The irregularity α of f should be compensated by more integrability p for the data. If α is close or equal to 1, the condition p ≥ 2 is too strong but our arguments are not sufficient in the proof.
For Type-II BSVIEs (1.7) we are not able to provide a similar result as Theorem 3.10. In its proof, the generated troubles for p = 2 are pointed out. But let us detail this issue already.
The reason can be understood just by considering the term Z. Since Z is integrated w.r.t. the Brownian motion W in the BSVIE, the natural norm for Z is: But it is symmetric w.r.t. (t, s) only for p = 2. The two time variables t and s do not play the same role and the integrability property is not the same w.r.t. t or w.r.t. s, except if p = 2. Thereby in BSVIE (1.4), we can use both Z(t, s) and Z(s, t) if p = 2 (Prop. 3.12). Let us also mention that in the case where the generator depends on the stochastic integrand w.r.t. a Poisson random measure, the case when p < 2 has to be handled carefully. Indeed in this case, Burkholder-Davis-Gundy inequality with p/2 < 1 does not apply and the L p/2 -norm of the predictable projection cannot be controlled by the L p/2 -norm of the quadratic variation (see [25] and the discussion in [24]). The extension to p = 2 seems difficult to prove and is left for further research. We also stress that p ≥ 2 implies that L p (S, T ; H p (R, T )) ⊂ L 2 (S, T ; H 2 (R, T )) and L p (S, T ; L p π (R, T )) ⊂ L 2 (S, T ; L 2 π (R, T )): For 1 < p < 2, this property fails. However for p = 2, coming back to the BSVIE (1.7), B t = t and if (H3*) holds, then 1 ≤ e βAt ≤ e βKT .

Existence and uniqueness for BSVIEs (1.4) and (1.5)
The aim of this section is to prove Theorems 3.3 and 3.9.

Preliminary results
which is a special case of (1.5) where f does not depend on y. Using (3.1), this BSVIE can be rewritten for any t ∈ [S, T ] Let us prove the next result.
Proof. Note that for any t ∈ [0, T ] Arguing as in the proof of Lemma 3.8 in [31], we derive that for any γ > 0 and δ > 0 T t e δAs |h(t, s, Z(t, s), U (t, s))| 2 α 2 s dB s dA t .
From our assumption (H2) on h, we deduce: thus from the definition of α (hypothesis (H3)) e δAr |h(t, r, Z(t, r), U (t, r))| 2 α 2 Thereby for any t ∈ [S, T ] Hence we obtain that This leads to the next estimate on H: Now remark that Thus for γ > 0 and δ > 0 Note that we used many times that A t is F u -measurable and Corollary D.1 in [31], together with (4.7) for the last inequality. For γ > δ, The same holds for H(u), thus For the second term in (4.9) Combining (4.6), (4.8) and the previous estimate, we deduce for any δ < γ ≤ β: If f and β are such that M f (β) < 1/2, then the conclusion of the lemma follows.
Proof. We outline the proof of Proposition 3.13 in [31]. Note that If dh(t, r) = h(t, r, Z(t, r), U (t, r)) −h(t, r, Z(t, r), U (t, r)), and from the assumptions onh we obtain: As for the previous lemma, we deduce: Thereby for any δ < γ ≤ β: The conclusion of the lemma follows as for the previous lemma.
Note that this stability result leads to the uniqueness of the solution of BSVIE (4.4) in the space S 2 δ * (β)≤β (∆ c (0, T )). The following result can be now stated:

Type-II BSVIE (1.4) for deterministic B
The aim of this section is the study of the Type-II BSVIE (1.4)

Let us emphasize that the first equality only holds because
we deduce that Let us consider space S 2 δ≤γ (R, T ), the set of all (y, z, u, m) in S 2 δ≤γ (R, T ) such that for dB-a.e. t ∈ [R, T ] a.s.

More properties in the Itô framework
In this section, the setting developed in Sections 2.2 and 3.3 is used, and the goal is to prove Theorems 3.10 and 3.11, which are extensions of some results of [46].
In this case, space S 2 (∆ c (0, T )) can be more easily defined here and the notations of Sections 2.1, 2.2 and 3.1 (essentially β = 0) are adapted, which leads to the same notations as in [46]. Hence some details are skipped (see [46], Sect. 2.1 for interested readers). For any p, q in [0, +∞), H = R d or R d×k , and S ∈ [0, T ], We identify For p = 2, this space corresponds to L 2 0,F S (0, T ) of Section 4. When adaptiveness is required, the subscript F S is replaced by F. The above spaces are for the free term Φ(·) (for which F-adaptiveness is not required) and for Y (·) (for which F-adaptiveness is required). Sometimes the subscript P is also used if predictability is needed.
To control the martingale terms in the BSVIE, other spaces are introduced. For any p, q ≥ 1 Again for p = q = 2, this space is equal to set H 2,• 0≤0 of Section 3.1. Let us also consider the case: Process N belongs to L q (S, T ; H p (S, T )) if and only if ψ ∈ L q (S, T ; L p π (S, T )), namely ψ is in the set of all processes ψ : [S, T ] 2 × R m → R d such that for almost all t ∈ [S, T ], ψ(t, ·, ·) ∈ L p π (S, T ) verifies Let us emphasize that for p = q = 2, this space corresponds to H 2, 0≤0 in Section 3.1. If martingale M is defined by (3.1), due to the orthogonality of the components of M , for any p > 1, there exist two universal constants c p and C p such that And M belongs to L q (S, T ; H p (S, T )) if and only if the triplet (Z, T )). Finally the product spaces are denoted: with the naturally induced norm. Definitions 3.1 (adapted solutions) and 3.6 (M-solutions) remain unchanged, except that (Y, Z, U, M ) belongs to S p (0, T ) and condition (3.7) becomes: To complete this presentation, let us set out some facts concerning the Poisson integral. From the Burkholder-Davis-Gundy inequality (see [36], Thm. 48), for all p ∈ [1, ∞) there exist two universal constants c p and C p (not depending on M ) such that for any càdlàg F-martingale M (·) and for any T ≥ 0 In particular (5.1) means that the Poisson martingale N is well-defined on [0, T ] (see Chapter II in [18]) provided we can control the expectation of N p/2 T for some p ≥ 1. From the Bichteler-Jacod inequality (see for example [29]), the two cases are distinguished: p ≥ 2 and p < 2.
Hence the generator of our BSVIE can be defined on Moreover ψ t is also in L 1 µ + L 2 µ . Thereby for p < 2, our generator is be defined on L 1 µ + L 2 µ (for the definition of the sum of two Banach spaces, see for example [22]).
See Section 1 of [24] for details on this point. In particular for N defined by if p ≥ 2, there exist two universal constants κ p and K p such that But if 1 < p < 2, there only exists a universal constant K p,T such that And it holds that L p µ + L 2 µ ⊂ L 1 µ + L 2 µ .

Proof of Theorem 3.10
Bear in mind that the aim is the proof of existence and uniqueness in space S p (0, T ) of the adapted solution (Y, Z, U, M ) of BSVIE (1.6): The proof is based on intermediate results. We consider BSVIE (4.4) which becomes here Then there exists a constant C depending on p, K and T , such that for t ∈ [S, T ] |h(t, r, Z(t, r), U (t, r)) −h(t, r, Z(t, r), U (t, r))| dr p .

(5.7)
This lemma is a consequence of Proposition 2.7 applied to the parametrized BSDE (4.3). The arguments are similar to those used for Lemmata 4.1 and 4.2 or for Corollary 3.6 of [46]. For completeness the main ideas are evoked here. The parametrized BSDE (4.3) becomes: From Proposition 2.7, for any Φ(·) ∈ L p F T (S, T ) the previous BSDE has a unique solution (λ(t, ·), z(t, ·), u(t, ·), m(t, ·)) in D p (R, T ) and for a.
Then there exists a constant C depending on p, K and T , such that |h(t, r, z(t, r), u(t, r)) −h(t, r, z(t, r), u(t, r))| dr (5.10) Note that here ψ S (t) is only required to be F S -measurable for almost all t and not F-adapted. Let us now proceed with the proof of Theorem 3.10.
Proof. The outline of the proof of Theorem 3.7 in [46] is followed.
Step 1. For any S ∈ [0, T ], let us consider the set S p (S, T ), the space of all (y, z, u, m) in S p (S, T ) such that for a.e. t ∈ [S, T ] a.s.  is bounded from above by CE|y(t)| p . On S p (S, T ) the following norm is considered: The same arguments as inequality (3.48) of [46] show the norm equivalence: (y, z, u, m) p If Φ ∈ L p F T (S, T ) and (y, ζ, ν, m) ∈ S p (S, T ), we consider the BSVIE on [S, T ]: dM (t, s).   . Note that at this step, (H3*) can be replaced by a weaker condition as in [46].
Let us take a short break in the proof to understand the trouble in the case of a Type-II BSVIE. The driver of the BSVIE (5.12) would be replaced by T t f (t, s, y(s), Z(t, s), ζ(s, t), U (t, s), ν(s, t)) ds.
Hence in (5.13), the next additional terms would appear: There are also other extra terms in (5.14). To circumvent this issue, this term could be added in the definition of the norm. However we cannot control this symmetrized version of the norm for (Z, U ) in this step, but also in the next one. In other words the map Θ is no longer a contraction.
Hence using (5.2) for p ≥ 2 or (5.3) for p < 2, and (5.13) and (5.15) yield to: It is a BSVIE with terminal condition ψ S ∈ L p F S (R, S) and generator f . As in the first step, this BSVIE has a unique solution in S p (R, S) provided that S − R > 0 is small enough. Now for t ∈ [R, S] from the expression (5.17) of ψ S , we obtain that Moreover using the same arguments as in the first step yields to: The stability result holds in our framework. The proof is based on the same arguments given in [46] (see Eq. (3.71) in particular) and is skipped here.

Time regularity (Thm. 3.11)
Let us describe several sets for the process Y .
Again when only measurability is required, subscript F is replaced by F S . If we want to deal with continuity, then D (resp. D) is changed to C (resp. C) (see [46], Sect. 2.1). Coming back to a generic martingale M (t, ·), the space Proof. The proof is an adaptation of the arguments of [44], together with [23,24], and is set out in the appendix (see Sect. A.3).
Now the proof of Theorem 3.11 is presented, following the outline of the proof of Theorem 2.4 in [44].
In particular a.s. t → Y (t) := X t (t) is càdlàg: Note that Using the BDG inequality (p ≥ 2), we have Recall that space H 2 is a Banach space (see [10] Step 2. Assume that f depends only on z and ψ. Let us define Z 0 (t, s) ≡ 0, U 0 (t, s) ≡ 0 and recursively for n ≥ 1: Arguing exactly as in [44] yields to: for any n ≥ 1 Let us now prove the convergence of Y n . Using Lemma 5.3, conditions (H2) and (H3*), we obtain Using inequality (5.2), taking β large enough (greater than 4(K 2 K 2 2 ), where K comes from (H3*) and K 2 from (5.2)) and iterating the previous inequality leads to: First taking the expectation and integrating w.r.t t ∈ [0, T ], the convergence of (Z n , U n , M n ) in H 2 is deduced. Then T t e βs |Z 1 (t, s)| 2 + U 1 (t, s, ·) 2 L 2 µ ds.
From (5.19), E(ξ p/2 ) < +∞ and t → E Ft (ξ) is a martingale. By Doob's maximal inequality where constant C does not depend on n. Thus there exists a càdlàg adapted process Y such that As an immediate consequence, the limit is the unique solution in S 2 (0, T ) of the BSVIE Step 3. Assume that f now also depends on y. Let us define Y 0 (t) ≡ 0 and for n ≥ 1: We know that t → Y n (t) is càdlàg. Using Lemma 5.3 again, we obtain: Thus for β = 8K 2 K 2 2 (again K 2 coming from (5.2)): Then By Doob's maximal inequality Since p ≥ 2, by Jensen's inequality, Define Arguing as above, with Fatou's lemma and the previous uniform (in n and s) estimate, we get Hence there is a càdlàg adapted process Y such that And from the above estimate, E sup |Y (s)| p < ∞. This achieves the proof of Theorem 3.11.

Existence and uniqueness for the Type-II BSVIE (1.7)
Coming back to BSVIE (1.7) and Proposition 3.12, suppose that Φ ∈ L 2 F T (0, T ), that (H2), (H3*) and (3.16) hold 7 : Nonetheless a direct proof could be given, following the outline of the proof of Theorem 3.10, that is of Theorem 3.7 in [46]. The modifications are quite obvious. In Step 1, fix Φ ∈ L 2 F T (S, T ) and (y, ζ, ν, m) ∈ S 2 (S, T ) and consider the BSVIE on [S, T ] The second step remains unchanged, whereas in the third step, Lemma 5.2 is used, with f S (t, s, z, u) = f (t, s, Y (s), z, Z(s, t), u, U (s, t)), (t, s, z, u) The last two steps are almost the same; the modifications are straightforward.

Comparison principle
In this section, dimension d is equal to one. Our goal is to extend some results contained in [42]. Note that the comparison principle for BSDEs has been proved in [23,24] in the quasi left-continuous case (see also [11], Thm. 3.2.1 or [34], Prop. 5.32). In Theorem 3.25 of [31], the comparison principle is established for the BSDE: Compared to BSDE (1.2), the difference is that f depends on Y s− , instead of Y s . This property is crucial in [31], since they have to take into account the discontinuity of B. Before stating the comparison principle, let us recall that a generator f can be "linearized" as follows: f (ω, s, y, z, u s (ω; ·) − f (ω, s, y , z , u s (ω; ·) = λ s (ω)(y − y ) + η s (ω)b s (ω)(z − z ) + f (ω, s, y, z, u(·)) − f (ω, s, y, z, u (·)).
See Remark 3.24 of [31]. Let us emphasize that λ and η also depend on y, y , z, z , u, u , c. In particular λ and η are not predictable if they depend on Y s . This is the reason why the previous BSDE (and not BSDE (1.2)) is studied for the comparison property in [31]. Nonetheless to simplify the notations and when no confusion may arise, we omit this dependence. If (F2) holds, |λ s (ω)| 2 ≤ s (ω) and |η s (ω)| 2 ≤ θ • s (ω) dP ⊗ dB-a.e. on Ω × [0, T ]. The comparison principle is the following (similar to [31], Thm. 3.25).
Driverf also verifies (H2)-(H3) and y →f (t, s, y, z, u) is non-decreasing. If a.s. for 0 ≤ t ≤ T , Φ 2 (t) ≥ Φ 1 (t), then the corresponding solutions of BSVIEs (1.5) with generator f i , verify for any t ∈ [0, T ]: Proof. Since the arguments of the proof are almost the same as in [42], the details are referred to the appendix.
If drivers f 1 and f 2 cannot be "separated" by a non-decreasing generatorf , the restriction to half-linear generators is introduced as in Theorem 3.9 of [42]. Hence suppose that generator f is linear w.r.t. z and ψ: f (t, s, y, z, u) = g(t, s, y) + h(s)b s z + K s (u(·)κ s (·)), (6.2) where h is a process bounded by θ • and κ : Ω × [0, T ] × R m → R is progressively measurable and such that (P2) holds. Our comparison result is an extension to the jump case of Theorems 3.8 and 3.9 of [42] for BSVIE (1.5) where f is given by (6.2). The main difference comes from the free terms. Indeed in [42] where B t = t, free term Φ is supposed to be in C F T ([0, T ], L 2 (Ω)) (see the functional spaces defined in Sect. 5.2). Hence for any Π is the mesh of the partition. In our setting, we cannot separate t and Ω, since B is random.

Now consider the terminal condition
By our conditions, ξ N is non-negative and thus a. for (t, s) ∈ (t N −2 , t N −1 ] × (t N −1 , t N ] and our previous BSVIE becomes Again we solve the BSDE: on [t N −2 , t N −1 ] and by the uniqueness and the comparison principle for BSDE, we deduce that By induction we obtain that Y Π (t) ≥ 0, t ∈ [0, T ]. From Theorem 3.3, the stability estimate for BSVIE yields to:  The conclusion of the proposition follows and this achieves the proof.

Comparison principle in the Itô setting
In the framework of Section 3.3, the results of Propositions 6.3 and 6.4 remain true here. Condition (6.7) holds in this case if free terms Φ i belong to C([0, T ], L 2 (Ω)), as in [42].
Let us emphasize that the role of the càdlàg property of X is important here. Thus it should be possible to relax the regularity assumption on the coeffcients Υ i or Ξ of the FSVIE. But as for a BSVIE, the regularity of the paths of X is neither a direct property nor an easy stuff.
Note that the extension of the duality result to the setting of Section 4 is an issue. Indeed a solution for the Type-II BSVIE (1.4) is first needed. But the orthogonality between B, X • , π and M is much more delicate and several simplifications are not true anymore with these processes.
With the previous duality result, it is possible to extend the comparison principle for M-solution of a Type-II BSVIE of the form: Up to some technical conditions, one can follow the scheme of Theorems 3.12 and 3.13 in [42].

Appendix A.
A.1 Proof of Lemma 3.2 Recall that for δ < γ ≤ β: The lemma states that the infimum of Π f (γ, δ) over all δ < γ ≤ β is given by M f (β) = Π f (β, δ * (β)), where δ * (β) is the unique solution on (0, β) of the equation: where λ is a one-dimensional predictable process s.t. |λ s (ω)| ≤ 1 s (ω) and η is a m-dimensional predictable process such that |η| 2 ≤ θ 1,• , dP ⊗ dB-a.e. (using (H2) for f 1 ). Let us choose γ = λ and from (P2) Now by the Girsanov transform, if Q is the probability measure defined by: where E(v) stands for the stochastic exponential operator, then the assumptions on κ imply that Q is equivalent to P. Taking the conditional expectation under Q in the previous inequality yields to The details concerning the disappeared martingale terms can be found in the proof of Theorem 3.25 in [31]. Hence the conclusion of the Proposition follows.
From this lemma, it is possible to deduce that Y belongs to D([0, T ]; L p F (Ω)), provided that we have regularity assumption on t → Φ(t) and t → f (t, s, y, z, ψ), as in Theorem 4.2 of [46] in the continuous setting. Note that the estimate on (Z, U, M ) derived before Lemma A.1 is crucial here. Let us emphasize again that it does not mean that Y is in D ([0, T ]; L p F (Ω)); in other words we do not deduce that a.s. the paths are càdlàg.
Let us now prove Lemma 5.3.