A unified approach to well-posedness of type-I backward stochastic Volterra integral equations

We study a novel general class of multidimensional type-I backward stochastic Volterra integral equations. Toward this goal, we introduce an infinite dimensional system of standard backward SDEs and establish its well-posedness, and we show that it is equivalent to that of a type-I backward stochastic Volterra integral equation. We also establish a representation formula in terms of non-linear semilinear partial differential equation of Hamilton-Jacobi-Bellman type. As an application, we consider the study of time-inconsistent stochastic control from a game-theoretic point of view. We show the equivalence of two current approaches to this problem from both a probabilistic and an analytic point of view.


Introduction
This paper is concerned with introducing a unified method to address the wellposedness of backward stochastic Volterra integral equations, BSVIEs for short. BSVIEs are regarded as natural extensions of backward stochastic differential equations, BS-DEs for short. On a complete filtered probability space (Ω, G, G, P), supporting an n-dimensional Brownian motion B, and denoting by G the P-augmented natural filtration generated by B, one is given data, that is to say a G T -measurable random variable ξ, and a mapping g, referred to respectively as the terminal condition and the generator. A solution to a BSDE is a pair of G-adapted processes (Y · , Z · ) such that Y t = ξ + T t g r (Y r , Z r )dr − T t Z r dB r , t ∈ [0, T ], P−a.s. (1.1) BSDEs of linear type were first introduced by Bismut [10,11] as an adjoint equation in the Pontryagin stochastic maximum principle. Actually, the contemporary work of Davis and Varaiya [20] 1 studied a precursor of a linear BSDE for characterising the value function and the optimal controls of stochastic control problems with drift control only. In the same context of the stochastic maximum principle, BSDEs of linear type are present in Arkin and Saksonov [7], Bensoussan [9] and Kabanov [33]. Remarkably, the extension to the non-linear case is due to Bismut [12], as a type of Riccati equation, as well as Chitashvili [16], and Chitashvili and Mania [17,18]. Later, the seminal work of Pardoux and Peng [40] presented the first systematic treatment of BSDEs in the general nonlinear case, while the celebrated survey paper of El Karoui, Peng, and Quenez [25] collected a wide range of properties and applications of BSDEs to finance. Among such properties we recall the so-called flow property, that is to say, for any 0 ≤ r ≤ T , Y t (T, ξ) = Y t (r, Y r (T, ξ)), t ∈ [0, r], P−a.s., and, Z t (T, ξ) = Z t (r, Y r (T, ξ)), dt ⊗ dP−a.e. on [0, r] × Ω, where (Y (T, ξ), Z(T, ξ)) denotes the solution to the BSDE with terminal condition ξ and final time horizon T .
A natural extension of (1.1) arises by considering a collection of G T -measurable random variables (ξ(t)) t∈[0,T ] , referred in the literature of BSVIEs as the free term, as well as a generator g. In such a setting, a solution to a BSVIE is a pair (Y · , Z · · ) of processes such that Y t = ξ(t) +    2) are referred in the literature as type-I and type-II BSVIEs, respectively. The first mention of such equations is, to the best of our knowledge, due to Hu and Peng [31]. Indeed, in the context of well-posedness of BSDEs valued in a Hilbert space, a prototype of type-I BSVIEs (1.3) is considered, see the comments following [31, Remark 1.1]. Two decades passed before a direct consideration of BSVIEs of the form A unified approach to well-posedness of type-I BSVIEs [29,Lemma A.2.3]. Although following different approaches, their analyses lead to introduce type-I BSVIEs of the form Y t = ξ(t) + T t g r (t, Y r , Z t r , Z r r )dr − T t Z t r dB r , P−a.s., t ∈ [0, T ]. (1.4) These are BSVIEs in which the diagonal of Z appears in the generator. We highlight that, until the present work, the only well-posedness results in the literature for type-I BSVIEs (1.4) are available in [54] and [29]. Both results hold for the particular case in which the driver g is linear in Z t r . Indeed, the argument in [54] follows as a consequence of the representation formula, i.e. an analytic argument via PDEs, and holds in a Markovian setting. On the other hand, the probabilistic argument in [29] holds in the non-Markovian case.
Likewise, Hamaguchi [26,27] studied a time-inconsistent control problem where the cost functional is defined by the Y component of the solution of a type-I BSVIE (1.3), in which g depends on a control. Via Pontryagin's optimal principle, the author noticed that the adjoint equations correspond to an extended type-I BSVIE, as first introduced in Wang [53] in the context of generalising the celebrated Feynman-Kac formula. An extended type-I BSVIE consists of a pair (Y · · , Z · · ), with appropriate integrability, such that s −→ Y s is continuous in an appropriate sense for s ∈ [0, T ], Y s · is pathwise continuous, Z s · is predictable, and Y s t = ξ(s) + (1. 5) We highlight that the noticeable feature of (1.4) and (1.5) is the appearance of the 'diagonal' processes (Y t t ) t∈[0,T ] and (Z t t ) t∈[0,T ] , respectively. A prerequisite for rigorously introducing these processes is some regularity of the solution. Indeed, the regularity of s −→ (Y s , Z s ) in combination with the pathwise continuity of Y and the introduction of a derivative of Z s with respect to s, as first discussed in [29], make the analysis possible, see Remark 3.4 for details.
Put succinctly, type-I BSVIEs, understood in a broader sense than that of (1.3), provide a rich framework to address new classes of problems in mathematical finance and control. In the case of time-inconsistent control problems, (1.4) and (1.5) appear as a consequence of the study of such problems via Bellman's and Pontryagin's principles, respectively. Consequently, in this paper we want to build upon the strategy devised in [29] and address the well-posedness of a general and novel class of type-I BSVIEs. We let X be the solution to a drift-less stochastic differential equation (SDE, for short) under a probability measure P, and F be the P-augmentation of the filtration generated by X, see Section 2.1 for details, and consider a tuple (Y · · , Z · · , N · · ), of appropriately F-adapted processes, which for any s ∈ [0, T ] satisfy, P−a.s. for any t ∈ [0, T ], the equation We remark that the additional process N corresponds to a martingale process which is P-orthogonal to X. This is a consequence of the fact that we work with a general filtration F. To the best of our knowledge, a theory for type-I BSVIEs, as general as the ones introduced above, remains absent in the literature. Moreover, such class of type-I BSVIEs has only been mentioned in [27,Remark 3.8] as an interesting generalisation of (1.5).
Our approach is based on the following class of infinite families of BSDEs, given for EJP 26 (2021), paper 89.
where (Y, Z, N , Y, Z, N ) are unknown, and required to have appropriate integrability, see Section 3 and Equation (S). We first establish the well-posedness of (S), see Theorem 3.6. For this it is important to be able to identify the proper spaces to carry out the analyses, see Remark 3.4. Moreover, we show that, for an appropriate choice of data for (S), its well-posedness is equivalent to that of the type-I BSVIE (1.6), see Theorem 4.4. Noticeably, our approach can naturally be specialised to obtain the well-posedness of (1.3), (1.4) and (1.5) in the classic spaces, see Remark 4.5. Moreover, as our results provide an alternative approach to BSVIEs, it may allow for the future design of new numerical schemes to solve type-I BSVIEs, which to the best of our knowledge, remain limited to [8]. In addition, we recover classical results for this general class of multidimensional type-I BSVIEs. We provide a priori estimates, show the stability of solutions as well as a representation formula in terms of a semilinear PDEs, see Proposition 5.1. Given our multidimensional setting, we refrained from considering comparison results, see Wang and Yong [63] for the one-dimensional case.
As an application of our results, we consider the game-theoretic approach to timeinconsistent stochastic control problems. We recall this approach studies the problem faced by the, so-called, sophisticated agent who aware of the inconsistency of its preferences seeks for consistent plans, i.e. equilibria. We show that as a consequence of Theorem 4.4, one can reconcile two recent probabilistic approaches to this problem. Moreover, we provide, see Proposition 5.3, an equivalent result for two earlier analytic approaches, based on semi-linear PDEs. We believe this helps to elucidate connections between the different takes on the problem available in the literature.
The rest of the paper is structured as follows. Section 2 introduces the stochastic basis on a canonical space as well as the integrability spaces necessary to our analysis. Section 3 precisely formulates the class of infinite families of BSDEs (S), which is the crux of our approach, and provides the statement of its well-posedness, while the proof is deferred to Section 6. Section 4 introduces the class of type-I BSVIEs which are the main object of this paper, and establishes the equivalence of its well-posedness with that of (S) for a particular choice of data. Section 5 deals with the representation formula for the class of type-I BSVIEs considered, and presents the application of our results in the context of time-inconsistent stochastic control. Finally, Section 6 includes the analysis of (S). For (Ω, F) a measurable space, Prob(Ω) denotes the collection of probability measures on (Ω, F). For a filtration F := (F t ) t∈[0,T ] on (Ω, F), P pred (E, F) (resp. P prog (E, F), P opt (E, F), P meas (E, F)) denotes the set of E-valued, F-predictable processes (resp. Fprogressively measurable processes, F-optional processes, F-adapted and measurable). For P ∈ Prob(Ω), F P := (F P t ) t∈[0,T ] , denotes the P-augmentation of F, where for t ∈ [0, T ], F P t := F t ∨ σ(N P ), where N P := {N ⊆ Ω : ∃B ∈ F, N ⊆ B and P[B] = 0}. With this, P ∈ Prob(Ω) can be extended so that (Ω, F, F P , P) becomes a complete probability space, see Karatzas and Shreve [34,Chapter II.7]. F P + denotes the right limit of F P , i.e. F P t+ := ε>0 F P t+ε , t ∈ [0, T ), and F P T + := F P T , so that F P + is the minimal filtration that contains F and satisfies the usual conditions.

The stochastic basis on the canonical space
We fix two positive integers n and m, which represent respectively the dimension of the martingale which will drive our equations, and the dimension of the Brownian motion appearing in the dynamics of the former. We consider the canonical space X := C([0, T ], R n ), with canonical process X. We let F be the Borel σ-algebra on X (for the topology of uniform convergence), and we denote by F o := (F o t ) t∈[0,T ] the natural filtration of X. We fix a bounded Borel measurable map σ : [0, T ] × X −→ R n×m , σ · (X) ∈ P meas (R n×m , F o ), and an initial condition x 0 ∈ R n . We assume there is P ∈ Prob(X ) such that P[X 0 = x 0 ] = 1 and X is martingale, whose quadratic variation, X = ( X t ) t∈[0,T ] , is absolutely continuous with respect to Lebesgue measure, with density given by σσ .
Enlarging the original probability space, see Stroock We now let F := (F t ) t∈[0,T ] be the (right-limit) of the P-augmentation of F o . We stress that we will not assume P is unique. In particular, the predictable martingale representation property for (F, P)-martingales in terms of stochastic integrals with respect to X might not hold. Remark 2.1. We remark that the previous formulation on the canonical is by no means necessary. Indeed, any probability space supporting a Brownian motion B and a process X satisfying the previous SDE will do, and this can be found whenever that equation has a weak solution.

Functional spaces and norms
We now introduce our spaces. In the following, (Ω, F T , F, P) is as in Section 2.1. We are given a finite-dimensional Euclidean space, i.e. E = R k for some non-negative integer k and | · | denotes the Euclidean norm. For any (p, q) ∈ (1, ∞) 2 , we introduce the EJP 26 (2021), paper 89.
• M p (E) of martingales M ∈ P opt (E, F), P-orthogonal to X (that is the product XM is an (F, P)-martingale), with P−a.s. càdlàg paths, M 0 = 0 and M p Finally, given an arbitrary integrability space (I p (E), · I ), we introduce the space Lastly we introduce the space, see Remark 2.2 for further details, ðZ r t dr, and, Z 2 H p,2 := Z 2 H 2,2 + Z 2 H 2 < ∞ Remark 2.2. When p = q, we will write L p (E) resp. L p,2 (E) for L q,p (E) resp. L q,p,2 (E) . With this convention, L 2 (E) resp. L 2,2 (E) will be L 2,2 (E) resp. L 2,2,2 (E) . Also, S p,2 (E), L q,p,2 (E) and H p,2 (E) are Banach spaces. In addition, we remark that the space H 2 (E) being closed implies H p,2 (E) is a closed subspace of H p,2 (E) and thus a Banach space. The space H p,2 (E) allows us to define a good candidate for (Z t t ) t∈[0,T ] as an element of H 2 (E). 4 Let Ω : so that the Radon-Nikodým property and Fubini's theorem imply 3 We recall that H 2 , being a Hilbert space and in particular a reflexive Banach space, has the so-called Radon-Nikodým property, see [

An infinite family of BSDEs
We are given jointly measurable mappings h, g, ξ and η, Moreover, we work under the following set of assumptions.

Remark 3.2.
We comment on the set of requirements in Assumption 3.1. Of particular interest is Assumption 3.1.(i), the other being the standard Lipschitz assumptions on the generators as well as their integrability at zero. Anticipating the introduction of (S) below and the discussion in Remark 3.4, Assumption 3.1.(i) will allow us to identify the second BSDE in the system as the antiderivative of the third one, see Remark 3.4.

Remark 3.4.
We now expound on our choice for the set-up and the structure of (S).
(i) We first highlight two aspects which are crucial to establish the connection between (S) and type-I BSVIE (1.6). The first is the presence of ∂U in the generator of the first equation. This causes the system to be fully coupled but is nevertheless necessary in our methodology, this will be clear from the proof of Theorem 4.4 in Section 4. The second relates to our choice to write three equations instead of two. In fact, our approach is based on being able to identify ∂U as the derivative with respect to the s variable of U in an appropriate sense and, at least formally, it is clear that the third equation allows us to do so, see Lemma 6.1 for details. Alternatively, we could have chosen not to write the third equation and consider for any s ∈ [0, T ], the system, which holds P−a.s. for any t ∈ [0, T ], where d ds U s corresponds to the density with respect to the Lebesgue measure of s −→ U s .
Nevertheless, for the proof of well-posedness of (S) that we present in Section 6, we have to derive appropriate estimates for (∂U t t ) t∈[0,T ] , and for this it is advantageous to do the identification by adding the third equation in (S) and work on the space (H, · H ).
(ii) We also emphasise that the presence of (V t t ) t∈[0,T ] in the generator of the first equation requires us to reduce the space of the solution from the classic (H, and · H denotes the norm induced by H. Ultimately, this is due to the presence of (Z t t ) t∈[0,T ] in the type-I BSVIE (1.6). On this matter, we stress that to the best of our knowledge, our results constitute the first comprehensive study of type-I BSVIEs as general as (1.6). We remark that our identification of the appropriate space to carry out the analysis is based on [27, Section 2.1]. In the case where (V t t ) t∈[0,T ] (resp. (Z t t ) t∈[0,T ] does not appear in the generator of the first BSDE in (S) (resp. type-I BSVIE (1.6)), Proposition 6.5 (resp. Remark 4.5) provide the arguments for how one can adapt our approach to yield a solution in the classical space. This shows that our methodology recovers existing results on type-I BSVIE (1.3) as well as the so-called extended type-I BSVIE (1.5).
EJP 26 (2021), paper 89. Remark 3.5. In addition, we highlight two features of (S) that will come into play in the setting of type-I BSVIE (1.6), and differ from the one in the classic literature. They are related to the fact we work under the general filtration F. The first is the fact that the stochastic integrals in (S) are with respect to the canonical process X. Recall that σ is not assumed to be invertible (it is not even a square matrix in general and can vanish), therefore the filtration generated by X is different from the one generated by B. This yields more general results and it allows for extra flexibility necessary in some applications, see [29] for an example. The second difference is the presence of the processes (N, M, ∂M ). As it was mentioned in Section 2.1, we work with a probability measure for which the martingale representation property for F-local martingales in terms of stochastic integrals with respect to X does not necessarily hold. Therefore, we need to allow for orthogonal martingales in the representation. Certainly, there are known properties which are equivalent to the orthogonal martingales vanishing, i.e. N = M = ∂M = 0, for example when P is an extremal point of the convex hull of the probability measures that satisfy the properties in Section 2.1, see [32,Theorem 4.29]. Assumption 3.1 provides an appropriate framework to derive the well-posedness of (S). The following is the main theorem of this section whose proof we postpone to Section 6.
where for ϕ ∈ {Y, Z, N , U, V, M, ∂U, ∂V, ∂M, ξ, η, ∂ s η} and Φ ∈ {h, g, ∇g} The reader may wonder about our choice to leave out the diagonal of ∂V in the generator of the first equation in (S). As we will argue below, this would require us to consider an auxiliary infinite family of quadratic BSDEs. Since the main purpose of this paper is to relate the well-posedness of (S) to that of the type-I BSVIE (1.6), and inasmuch as we do not need to consider this case to establish Theorem 4.4, we have refrained from pursuing it in this document. Nevertheless, this case is covered as part of the study of the extension of (S) to the quadratic case in Hernández [28]. If we were to study the system, which for any s ∈ [0, T ] satisfies to make sense of the family of BSDEs with terminal condition ∂ ss η and generator ∇ 2 g t (s, x,ũ,ṽ, u, v, u, v, y, z) := ∇g t (s, x,ũ,ṽ, u, v, y, z) where Π := s, u, v 1: , ..., v n: ,Π := 1, u, v 1: , ..., v n: and ∂ 2 πiπj g t (s, x, u, v, y, z) denote the second order derivatives of g. Even though we could add assumptions ensuring that the second order derivatives are bounded, it is clear from the second term in the generator that we would necessarily need to consider a quadratic framework.

Well-posedness of type-I BSVIEs
We now address the well-posedness of type-I BSVIEs. Let d be a non-negative integer, and f and ξ be jointly measurable functionals such that for any (s, y, z, u, v) To derive the main result in this section, we will exploit the well-posedness of (S). Therefore, we work under the following set of assumptions.
∂ z:i f t (s, x, y, z, u, v)v i: , satisfies ∇f · (s, ·, y, z, u, v, u, v) ∈ P prog (R d , F) for all s ∈ [0, T ]; (ii) for ϕ ∈ {f, ∂ s f }, (u, v, y, z) −→ ϕ t (s, x, y, z, u, v) is uniformly Lipschitz continuous, i.e. ∃L ϕ > 0, such that for all (s, t, x, y,ỹ, z,z, u,ũ, v,ṽ) |ϕ t (s, x, y, z, u, v) − ϕ t (s, x,ỹ,z,ũ,ṽ)| We consider the n-dimensional type-I BSVIE on (H , · H ), which for any s ∈ [0, T ] holds P−a.s. for any t ∈ [0, T ], We work under the following notion of solution.  Defining h t (x, y, z, u, v, u) := f t (t, x, y, z, u, v) − u, we may consider the system, given for any s ∈ [0, T ] by In particular, we highlight that type-I BSVIE (1.5), in which the diagonal of Y , but not of Z is allowed in the generator, had been considered in [27; 53]. In such a scenario, the authors assumed (ξ, f ) ∈ L 2,2 (R d ) × L 1,2,2 (R d ), and no additional condition is required to obtain the well-posedness of (1.5). As it will be clear from Proposition 6.5 and Remark 4.5 our procedure can be adapted to work under such set of assumptions provided the diagonal of Z is not considered in the generator.
(ii) Moreover, the spaces of the solution considered in [27; 53] also differ, echoing the absence of the diagonal of Z in the generator. The authors work with the notion of C-solution, that is, Y is assumed to be a jointly measurable process, such that s −→ Y s is continuous in L 1,p (R d ), p ≥ 2, and for every s ∈ [0, T ], Y s is F-adapted with P−a.s. continuous paths. This coincides with our definition of the space L 1,p,2 (R d ). Similarly, Z belongs to the space H 2,2 (R n×d ). On the other hand, [54] provides a representation formula for type-I BSVIEs for which the driver allows for the diagonal of Z, but not of Y .
More precisely, they introduce a PDE, similar to the one we will introduce in Section 5, prove its well-posedness, and then a Feynman-Kac formula. Naturally, in this case (Y, Z) inherits the regularity of the underlying PDE.
(iii) The main contribution of our methodology to the field of BSVIEs is to be able to accommodate type-I BSVIEs for which the diagonal of Z appears in the generator. For this, the definition of the space (H , · H ), notably the space H 2,2 (R n×d ), and (v) Let us remark that Assumption 4.1.(i), being an assumption on the data of the BSVIE, is easier to verify in practice compare to the regularity required in [27]. Certainly, our results would still hold true if we require the differentiability of data (ξ, f ) with respect to the parameter s in the L 2 (resp. L 1,2 ) sense, or, even better, absolute continuity,.
(vi) Lastly, we stress that the above type-I BSVIE is defined for (s, t) ∈ [0, T ] 2 , as opposed to 0 ≤ s ≤ t ≤ T . However, anticipating the result of Theorem 4.4, this could be handled by first solving on (s, t) ∈ [0, T ] 2 and then consider the restriction to 0 ≤ s ≤ t ≤ T .
We are now in position to prove the main result of this paper. The next result shows that under the previous choice of data for (S f ), its solution solves the type-I BSVIE with data (ξ, f ) and vice versa.  (ii) the type-I BSVIE (4.1) is well-posed, and for any (Y, Z, N ) ∈ H solution to type-I BSVIE (4.1) there exists C > 0 such that Proof. (ii) is a consequence of (i). Indeed, (4.2) follows from Proposition 6.3, and the well-posedness of type-I BSVIE (4.1) from that of (S f ), which holds by Assumption 4.1 and Theorem 3.6. We now argue (i). Let (Y, Z, N , Y, Z, N, ∂Y, ∂Z, ∂N ) ∈ H be a solution to (S f ). It then follows from Lemma 6.2 that, P−a.s. for any t ∈ [0, T ], where N t := N t t − t 0 ∂N r r dr, t ∈ [0, T ], and N ∈ M 2 (R d ). This shows that solves the first BSDE in (S f ). It then follows from the well-posedness of (S f ), which holds by Assumption 4.1 and Theorem 3.
We are left to show the converse result. Let (Y, Z, N ) ∈ H be a solution to type-I BSVIE (4.1). We begin by noticing that the processes Y : EJP 26 (2021), paper 89.
and Assumption 4.1 holds, we can apply Lemma 6.1 and obtain the existence of (∂Y, ∂Z, ∂N ) ∈ S 2,2 (R d )×H 2,2 (R n×d )×M 2,2 (R d ) such that for s ∈ [0, T ], P−a.s. for t ∈ [0, T ], For this, we first note that in light of Lemmata 6.1 and 6.2 we have that (4.4) and N ∈ M 2,2 (R d ). We are only left to argue Y ∈ S 2 (R d ). Note that by Assumption 4.1 and Equation (6.1) there exists C > 0 such that We conclude h H < ∞, h ∈ H and thus h solves (S f ).

Type-I BSVIEs, parabolic PDEs and time-inconsistent control
This section is devoted to the application of our results in Section 4 to the problem of time-inconsistent control for sophisticated agents. Moreover, we also reconcile seemingly different approaches to the study of this problem.

Representation formula for adapted solutions of type-I BSVIEs
Building upon the fact that second-order, parabolic, semilinear PDEs of HJB type admit a non-linear Feynman-Kac representation formula, we can identify the family of PDEs associated to Type-I BSVIEs. This is similar to the representation of forward backward stochastic differential equations, FBSDEs for short, see [64].
Then, Y s t := V(s, t, X t ), and Z s t := ∂ x V(s, t, X t ) define a solution to the type-I BSVIE given for every s ∈ [0, T ] by Proof. Let s ∈ [0, T ] and P as in Section 2. Applying Itō's formula to the process Y s t we find that P−a.s.
We verify the integrability of (Y, Z). As σ is bounded, X t has exponential moments of any order which are bounded on [0, T ], i.e. ∃C > 0, such that sup t∈[0,T ] E P [exp(c|X t | 1 )] ≤ C < ∞, for any c > 0, where C depends on T and the bound on σ. This together with the growth condition on V(s, t, x) and ∂ x V(s, t, x) yield the integrability.

Remark 5.2.
In the previous result the type-I BSVIEs has an additional term linear in z. This is a consequence of the dynamics of X under P, see Section 2.1. Nevertheless, as b is bounded, we can define P b ∈ Prob(X ), equivalent to P so that by Girsanov's theorem B b := B − · 0 b r (X r )dr is a P b -Brownian motion and r , t ∈ [0, T ], P b −a.s., 5 Here, | · | 1 denotes the 1 norm in R n , i.e. for x ∈ R n , |x| 1 := d i=1 |x i | EJP 26 (2021), paper 89.

On equilibria and their value function in time-inconsistent control problems
The game theoretic approach to time-inconsistent control problems in continuoustime started with the Markovian setting, and is grounded in the notion of equilibrium first proposed in Ekeland and Pirvu [24], Ekeland and Lazrak [23], and the infinite family of PDEs, or Hamilton-Jacobi-Bellman equation, provided by Björk, Khapko, and Murgoci [13], see Equation (5.2) below. Soon after, Wei, Yong, and Yu [68] presented a verification argument via a one dimensional PDE, but over an extended domain, see Equation (5.3) below. Both approaches have generated independent lines of research in the community, including both analytic and probabilistic methods, but no compelling connections have been established, as far as we know.
BSDEs and BSVIEs appear naturally as part of the probabilistic study of these problems. This approach allows extensions to a non-Markovian framework, and to reward functionals given by recursive utilities. Indeed, the approaches in [29] and [54] address these directions, and can be regarded as extensions of [13] and [68], respectively. As such, it is not surprising that in order to characterise an equilibrium and its associated value function, both [29] and [54] lay down an infinite family of BSDEs, and a type-I BSVIEs, respectively. In fact, [29,Theorem 3.7] and [54, Theorem 5.1] establish representation formulae for the analytic, i.e. PDEs, counterparts. Moreover, [29] noticed that their approach through BSDEs led to the well-posedness of a BSVIE. This is nothing but a manifestation of Theorem 4.4 which reconciles, at the probabilistic level, the findings of [29] and [54]. Moreover, we also include Proposition 5.3 which does the same at the PDE level. To sum up, we can visualise this in the next picture.
Following the approach of [54], let us assume that given an admissible A-valued strategy ν over the interval [s, T ], the reward at s ∈ [0, T ] is given by the value at s of the Y coordinate of the solution to the type-I BSVIE given by EJP 26 (2021), paper 89. [54] finds that the value along the equilibrium policy coincides with the Y coordinate of the following type-I BSVIE where the diagonal of Z appears in the generator. However, decoupling the dependence between the time variable and the variable source of time-inconsistency, we can define, It then follows from Theorem 4.4 that this approach is equivalent to that of [29] based on the system, which for any s ∈ [0, T ] holds P−a.s. for t ∈ [0, T ], We now move on to establish the connection of the analyses at the PDE level. The original result of [13] is based on the semi-linear PDE system of HJB type given for On the other hand, [68] considers the equilibrium HJB equation for J (s, t, x) ∈ C 1,1,2 ([0, T ] 2 × R n ) given by

Analysis of the BSDE system
Let us first recall the elementary inequalities valid for any positive integer n and any collection (a i ) 1≤i≤n of non-negative numbers, as well as, Young's inequality which guarantees that for any ε > 0, 2ab ≤ εa 2 + ε −1 b 2 .
In order to alleviate notations, and as it is standard in the literature, we suppress the dependence on ω, i.e. on X, in the functionals, and, write E instead of E P as the underlying probability measure P is fixed. Moreover, we will write I 2 instead of I 2 (E) for any of the integrability spaces involved, the specific space E is fixed and understood without ambiguity.

Regularity of the system and the diagonal processes
In preparation to the proof of Theorem 3.6, we present next a couple of lemmata from which we will benefit in the following. As a historical remark, we mention the following is in the spirit of the analysis in Protter [43,Section 3] and Pardoux and Protter [41] of forward Volterra integral equations.
A technical detail is to identify appropriate spaces so that given (∂U, U, V, M ) one can rigorously define the processes It is known that for U ∈ S 2,2 , the diagonal process (U t t ) t∈[0,T ] is well-defined. Indeed, this follows from the pathwise regularity of U s for s ∈ [0, T ] and has been noticed since [27; 29; 65]. The same argument works for (∂U, M ) ∈ S 2,2 × M 2,2 . Unfortunately, the same reasoning cannot be applied for arbitrary V ∈ H 2,2 and motives the introduction of the space H 2,2 , see Remark 3.4. T t e cr |U s r | 2 + |σ r V s r | 2 + |Y r | 2 + |σ r Z r | 2 dr ; (iv) V ∈ H 2,2 . Moreover, for V := (V t t ) t∈[0,T ] and ε > 0, P−a.s.
Proof. Note that in light of Assumption 3.1.(i), for (t, s, x, u, v, y, z) Therefore, for any s ∈ [0, T ] the second equation defines a linear BSDE, in (∂U s , ∂V s ), whose generator at zero, by Assumption 3.1.(iii), is in L 1,2 . Therefore, its solution (∂U s , ∂V s , ∂M s ) ∈ S 2 × H 2 × M 2 is well-defined from classic results, see for instance Zhang [73] or [25]. (ii) follows from classic a priori estimates, but when the norms are considered over where in the second inequality we exploited the fact (u, v, y, z) −→ ∂ s g(t, s, x, u, v, y, z) is Lipschitz, see Assumption 3.1.(iii), and C > 0 was appropriately updated. Next, we assume (iii) and show (iv). In light of (i) and (iii), s −→ ∂V s is the density of s −→ V s with respect to the Lebesgue measure. Arguing as in Remark 2.2, we obtain that we can define We now verify (6.5). By definition of V, Fubini's theorem and Young's inequality we have that for ε > 0 Thus V H 2 < ∞ and consequently V ∈ H 2,2 . This proves (iv). We now argue (iii). We also remark that a similar argument to the one in (i) shows that under Assumption 3.1 U ∈ S 2,2 . We know the mapping [0, T ] s −→ (∂Y s , ∂Z s , ∂M s ) EJP 26 (2021), paper 89. is continuous, in particular integrable. A formal integration with respect to s to (6.2) We then notice that for any t ∈ [0, T ] We now note that the integrability of (∂U To conclude, we first note that by our choice of (Π ) , I (∂U ) converges to the Lebesgue integral of ∂U s . In addition, the uniform continuity of s −→ ∂ s g(s, x, u, v, y, z) and s −→ ∂ s η(s, x), see Assumption 3.1.(i), justifies, via bounded convergence, the convergence in S 2,2 (resp. H 2,2 ) of I (∂U s ) to U T − U s (resp. I (∂V s ) to V T − V s ) as −→ 0. The result follows in virtue of the uniqueness of (U, V, M ).