Mean-Field Backward Stochastic Volterra Integral Equations

Mean-field backward stochastic Volterra integral equations (MF-BSVIEs, for short) are introduced and studied. Well-posedness of MF-BSVIEs in the sense of introduced adapted M-solutions is established. Two duality principles between linear mean-field (forward) stochastic Volterra integral equations (MF-FSVIEs, for short) and MF-BSVIEs are obtained. As applications, a multi-dimensional comparison theorem is proved for adapted M-solutions of MF-BSVIEs and a maximum principle is established for an optimal control of MF-FSVIEs.

On the other hand, a general (nonlinear) backward stochastic differential equation (BSDE, for short) introduced in Pardoux-Peng [28] is equivalent to the following: (1.8) Extending the above, the following general stochastic integral equation was introduced and studied in Yong [39,40,41]: Such an equation is called a backward stochastic Volterra integral equation (BSVIE, for short). A special case of (1.9) with g(·) independent of Z(s, t) and ψ(t) ≡ ξ was studied by Lin [23] and Aman-N'zi [3] a little earlier. Some relevant studies of (1.9) can be found in Wang-Zhang [37], Wang-Shi [36], Ren [31], and Anh-Grecksch-Yong [5]. Inspired by BSVIEs, it is very natural for us to introduce the following stochastic integral equation: where (Y (·), Z(· , ·)) is the pair of unknown processes, ψ(·) is a given free term which is F Tmeasurable (not necessarily lF-adapted), g(·) is a given mapping, called the generator, and Γ(t, s, Y, Z, Z) = lE θ(t, s, y, z,ẑ, Y, Z, Z) (y,z,ẑ)=(Y,Z, Z) (1.11) with (Y, Z, Z) being some random variables, for some mapping θ(·) (see the next section for precise meaning of the above). We call (1.10) a mean-field backward stochastic Volterra integral equation (MF-BSVIE, for short). Relevant to the current paper, let us mention that in Buckdahn-Djehiche-Li-Peng [9], mean-field backward stochastic differential equations (MF-BSDEs, for short) were introduced and in Buckdahn-Li-Peng [10] a class of nonlocal PDEs are studied with the help of an MF-BSDE and a McKean-Vlasov forward equation.
We see that MF-BSVIE (1.10) not only includes MF-BSDEs (which, of course, also includes standard BSDEs) introduced in [9,10], but also generalizes BSVIEs studied in [39,41,36], etc. in a natural way. Besides, investigating MF-BSVIEs allows us to meet the need in the study of optimal control for MF-FSVIEs. As a matter of fact, in the statement of Pontryagin type maximum principle for optimal control of a forward (deterministic or stochastic) control system, the adjoint equation of variational state equation is a corresponding (deterministic or stochastic) backward system, see [42] for the case of classical optimal control problems, [4,8,26] for the case of MF-FSDEs, and [39,41] for the case of FSVIEs. When the state equation is an MF-FSVIE, the adjoint equation will naturally be an MF-BSVIE. Hence the study of well-posedness for MF-BSVIEs is not avoidable when we want to study optimal control problems for MF-BSVIEs.
The novelty of this paper mainly contains the following: First, well-posedness of general MF-BSVIEs will be established. In doing that, we discover that the growth of the generator and the nonlocal term with respect to Z(s, t) plays a crucial role; a better understanding of which enables us to have found a neat way of treating term Z(s, t). Even for BSVIEs, our new method will significantly simplify the proof of well-posedness of the equation (comparing with [41]). Second, we establish two slightly different duality principles, one starts from linear MF-FSVIEs, and the other starts from linear MF-BSVIEs. We found that "Twice adjoint of a linear MF-FSVIE is itself", whereas, "Twice adjoint of a linear MF-BSVIE is not necessarily itself". Third, some comparison theorems will be established for MF-FSVIEs and MF-BSVIEs. It turns out that the situation is surprisingly different from the differential equation cases. Some mistakes found in [39,40] will be corrected. Finally, as an application of the duality principle for MF-FSVIEs, we establish a Pontryagin type maximum principle for an optimal control problem of MF-FSVIEs.
The rest of the paper is organized as follows. Section 2 is devoted to present some preliminary results. In Section 3, we prove the existence and uniqueness of adapted M-solutions to MF-BSVIE (1.10). In Section 4 we obtain duality principles. Comparison theorems will be presented in Section 5. In Section 6, we deduce a maximum principle of optimal controls for MF-FSVIEs.

Preliminary Results.
In this section, we will make some preliminaries.
Having some feeling about the operator Γ from the above, let us look at some useful properties of the operator Γ in general. To this end, we make the following assumption.
Definition 2.2. A pair of (Y (·), Z(· , ·)) ∈ H p [0, T ] is called an adapted M-solution of MF-BSVIE (1.10) if (1.10) is satisfied in the Itô sense and the following holds: It is clear that (2.9) implies This suggests us define M p [0, T ] as the set of all elements (y(·), z(· , ·)) ∈ H p [0, T ] satisfying: Note that for any (y(·), z(· , ·)) ∈ M 2 [0, T ], Relation (2.12) can be generalized a little bit more. To see this, let us present the following lemma. (2.13) Hereafter, K > 0 stands for a generic constant which can be different from line to line.
Proof. For fixed (S, t) ∈ ∆ (which means 0 ≤ S ≤ t ≤ T ) with S < t, let which is F t -measurable. Let (Y (·), Z(·)) be the adapted solution to the following BSDE: Then it is standard that Now, By taking conditional expectation lE[· | F S ], we see that Then (2.13) follows from (2.14).
We have the following interesting corollary for elements in M p [0, T ] (comparing with (2.12)).
From the above, we see that for any (y(·), z(·, ·)) ∈ M p [0, T ], and any β > 0, To conclude this subsection, we state the following corollary of Lemma 2.3 relevant to BSVIEs, whose proof is straightforward.
An lF-adapted process X(·) is called a solution to (2.19) if (2.19) is satisfied in the usual Itô sense. To guarantee the well-posedness of (2.19), let us make the following hypotheses.
Then a similar argument as above applies to obtain a unique solution of (2.31) on [δ, 2δ]. It is important to note that the step-length δ > 0 is uniform. Hence, by induction, we obtain the unique solvability of (2.19) on [0, T ].
Then we can obtain estimate (2.26).

Linear MF-FSVIEs and MF-BSVIEs.
Let us now look at linear MF-FSVIEs, by which we mean the following: (2.32) For such an equation, we introduce the following hypotheses.
(L1) The maps are measurable and uniformly bounded.
Mimicking the above, we see that general linear MF-BSVIE should take the following form: For the coefficients, we should adopt the following hypothesis.
We expect that under (L2), for reasonable ψ(·), the above (2.38) will have a unique adapted M-solution. Such a result will be a consequence of the main result of the next section.
In this section, we are going to establish the well-posedness of our MF-BSVIEs. To begin with, let us introduce the following hypothesis.
Note that we may take much more general g(·) and Γ(·). But the above is sufficient for our purpose, and by restricting such a case, we avoid stating a lengthy assumption similar to (H3) q . We now state and prove the following result concerning MF-BSVIE (3.4).
Note that the cases that we are interested in are p = 2, q. We will use them below.
Let us make some remarks on the above result, together with its proof.
We point out that even for the special case of BSVIEs, the proof we provided here significantly simplifies that given in [41]. The key is that we have a better understanding of the term Z(s, t) in the drift, and find a new way to treat it (see (3.16)). Now, let us look at linear MF-BSVIE (2.38). It is not hard to see that under (L2), we have (H3) q with q = 2. Hence, we have the following corollary.

Duality Principles.
In this section, we are going to establish two duality principles between linear MF-FSVIEs and linear MF-BSVIEs. Let us first consider the following linear MF-FSVIE (2.32) which is rewritten below (for convenience): (4.1) Let (L1) hold and ϕ(·) ∈ L 2 l F (0, T ; lR n ). Then by Corollary 2.7, (4.1) admits a unique solution X(·) ∈ L 2 l F (0, T ; lR n ). Now, let (Y (·), Z(· , ·)) ∈ M 2 [0, T ] be undetermined, and we observe the following: We now look at each term I i . First, for I 1 , we have Next, for I 2 , let us pay some extra attention on ω and ω ′ , Here, we have introduced the notation lE * , whose definition is obvious from the above, to distinguish lE (and lE ′ ). For I 3 , we have Finally, we look at I 4 .
Hence, we obtain On the other hand, suppose (L1) ′ holds and ϕ(·) ∈ C p l F ([0, T ]; lR n ). Then X(·) ∈ C p l F ([0, T ]; lR n ). Consequently, we obtain the following duality principle for MF-FSVIEs whose proof is clear from the above.
We call (4.2) the adjoint equation of (4.1). The above duality principle will be used in establishing Pontryagin's type maximum principle for optimal controls of MF-FSVIEs.
Next, different from the above, we want to start from the followng linear MF-BSVIE:  This is a special case of (2.38) in which Under (L2), by Corollary 3.4, for any ψ(·) ∈ L 2 l F (0, T ; lR n ), (4.4) admits a unique adapted M-solution (Y (·), Z(· , ·)) ∈ M 2 [0, T ]. We point out here that for each t ∈ [0, T ), the maps s →C 0 (t, s), s →C 1 (t, s) are lF-progressively measurable and lF 2 -progressively measurable on [t, T ], respectively. Now, we let a process X(·) ∈ L 2 l F (0, T ; lR n ) be undetermined, and make the following calculation: Similar to the above, we now look at the terms I i (i = 1, 2, 3, 4) one by one. First, we look at I 1 : Next, for I 2 , one has Now, for I 3 ,
We call MF-FSVIE (4.6) the adjoint equation of MF-BSVIE (4.4). Such a duality principle will be used to establish comparison theorems for MF-BSVIEs. Note that since for s < t,C 0 (s, t) T is F t -measurable and not necessarily F s -measurable, we have t ∈ (s, T ], (4.8) in general. Likewise, in general, We now make some comparison between Theorems 4.1 and 4.2.
First, we begin with linear MF-FSVIE (4.1) which is rewritten here for convenience: (4.10) According to Theorem 4.1, the adjoint equation of (4.10) is MF-BSVIE (4.2). Now, we want to use Theorem 4.2 to find the adjoint equation of (4.2) which is regarded as (4.4) with Then, by Theorem 4.2, we obtain the adjoint equation (4.6) with the coefficients: Hence, (4.10) is the adjoint equation of (4.2). Thus, we have the following conclusion: Twice adjoint equation of a linear MF-FSVIE is itself.
Next, we begin with linear MF-BSVIE (4.4). From Theorem 4.2, we know that the adjoint equation is linear MF-FSVIE (4.6). Now, we want to use Theorem 4.1 to find the adjoint equation of (4.6) which is regarded as (4.10) with Then by Theorem 4.2, the adjoint equation is given by (4.2) with coefficients: In another word, the twice adjoint equation of linear MF-BSVIE (4.4) is the following: which is different from (4.4), unlessC 0 (t, s) andC 1 (t, s) are F t -measurable for all (t, s) ∈ ∆. Thus, we have the following conclusion: Twice adjoint of a linear MF-BSVIE is not necessarily itself.

Comparison Theorems.
In this section, we are going to establish some comparison theorems for MF-FSVIEs and MF-BSVIEs, allowing the dimension to be larger than 1. Let When x ∈ lR n + , we also denote it by x ≥ 0, and say that x is nonnegative. By x ≤ 0 and x ≥ y (if x, y ∈ lR n ), we mean −x ≥ 0 and x − y ≥ 0, respectively. Moreover, if X(·) is a process, then by X(·) ≥ 0, we mean Also, X(·) is said to be nondecreasing if it is componentwise nondecreasing. Likewise, we may define X(·) ≤ 0 and X(·) ≥ Y (·) (if both X(·) and Y (·) are lR n -valued processes), and so on.
In what follows, we let e i ∈ lR n be the vector that the i-th entry is 1 and all other entries are zero. Also, we let Note that lM n×m + is the set of all (n × m) matrices with all the entries being nonnegative, lM n + is the set of all (n × n) matrices with all the off-diagonal entries being nonnegative, and lM n 0 is actually the set of all (n × n) diagonal matrices. Clearly, lM n + and lM n×m + are closed convex cones of lR n×n and lR n×m , respectively, and lM n 0 is a proper subspace of lR n×n . Whereas, for n = m = 1, one has lM 1 + = lM 1 0 = lR, lM We have the following simple result which will be useful below and whose proof is obvious.
In what follows, we will denote lM n + = lM n×n + .

Comparison of solutions to MF-FSVIEs.
In this subsection, we would like to discuss comparison of solutions to linear MF-FSVIEs. There are some positive and also negative results. To begin with, let us first present an example of MF-FSDEs.
Example 5.2. Consider the following one-dimensional linear MF-FSDE, written in the integral form: Taking expectation, we have Consequently, the solution X(·) is given by Thus, although X(0) = 1 > 0, the following fails: The above example shows that if the diffusion contains a nonlocal term in an MF-FSDE, we could not get an expected comparison of solutions, in general. Therefore, for linear MF-FSDEs, one had better only look at the following: with the diffusion does not contain a nonlocal term. For the above, we make the following assumption.
(C1) The maps are uniformly bounded, and they are lF-progressively measurable, and lF 2 -progressively measurable, respectively.
Note that, due to (5.1), the above (C1) is always true if n = 1. We now present the following comparison theorem for linear MF-FSDEs. Proof. It is known from Theorem 2.6 that as a special case of MF-FSVIE, the linear MF-FSDE (5.3) admits a unique solution X(·) ∈ L p l F (0, T ; lR n ) for any x ∈ lR n , and any p ≥ 2. Further, it is not hard to see that X(·) has continuous paths. Since the equation is linear, it suffices to show that x ≤ 0 implies To prove (5.6), we define a convex function where a + = max{a, 0} for any a ∈ lR. Applying Itô's formula to f (X(t)), we get We observe the following: (noting A 0 (s) ∈ lM n + ) Also, one has (making use of C 0 (s) ∈ lM Next, we have (noting A 1 (·) and f xx (·) are diagonal) Consequently, Hence, by Gronwall's inequality, we obtain Therefore, if x ≤ 0 (component-wise), then This leads to (5.6).
We now make some observations on condition (5.4).

Let
A 0 (·) = 0, A 1 (·) = 0, and C 0 (·) be continuous and for some i = j, i.e., at least one off-diagonal entry of C 0 (0) is negative. Then by a similar argument as above, we have that X(0) ≥ 0 does not imply X(t) ≥ 0.
The above observations show that, in some sense, conditions assumed in (5.4) are sharp for Proposition 5.3.
Based on the above, let us now consider the following linear MF-FSVIE: (5.7) Note that A 1 (·) is independent of t here. According to [34], we know that for (linear) FSVIEs (without the nonlocal term, i.e., C 0 (· , ·) = 0 in (5.7)), if the diffusion depends on both (t, s) and X(·), i.e., A 1 (t, s) really depends on (t, s), a comparison theorem will fail in general. Next, let us look at an example which is concerned with the free term ϕ(·).
Example 5.4. Consider the following one-dimensional FSVIE: for some b, σ ∈ lR. The above is equivalent to the following: The solution to the above is explicitly given by the following: We know that as long as σ = 0, for any t > 0 small and any K > 0, Therefore, we must have lP(X(t) < 0) > 0, ∀t > 0 (small).
The above example tells us that when σ = 0, or σ = 0 and b < 0, although the free term ϕ(t) = T − t is nonnegative on [0, T ], the solution X(·) of the FSVIE (5.7) does not necessarily remain nonnegative on [0, T ]. Consequently, nonnegativity of the free term is not enough for the solution of the MF-FSVIE to be nonnegative. Thus, besides the nonnegativity of the free term, some additional conditions are needed.
To present positive results, we introduce the following assumption. We now present the following result which is simple but will be useful later.
For the case that the diffusion is nonzero in the equation, we have the following result.
From the above proof, we see that one may replace b 0 (·) in conditions (5.19) by b 1 (·). Also, by an approximation argument, we may replace the derivatives in (5.19) of b 0 (·) and σ(·) by the corresponding difference quotients.

Comparison theorems for MF-BSVIEs.
In this subsection, we discuss comparison property for MF-BSVIEs. First, we consider the following linear MF-BSVIE: Note that Z(t, s) does not appear in the whole drift term, and Z(s, t) does not appear in the nonlocal term. Further, the coefficient of Z(s, t) is independent of s. Let us introduce the following assumption.
We have the following result.
In this section, we will briefly discuss a simple optimal control problem for MF-FSVIEs. This can be regarded as an application of Theorem 4.1, a duality principle for MF-FSVIEs. The main clue is similar to the relevant results presented in [39,41]. We will omit some detailed derivations.
General optimal control problems for MF-FSVIEs will be much more involved and we will present systematic results for that in our forthcoming publications.
For convenience, we make the following assumptions for the functions involved in the cost functional.
The purpose of presenting a simple optimal control problem of MF-FSVIEs here is to realize a major motivation of studying MF-BSVIEs. It is possible to discuss Bolza type cost functional. Also, some of the assumptions assumed in this section might be relaxed. However, we have no intention to have a full exploration of general optimal control problems for MF-FSVIEs in the current paper since such kind of general problems (even for FSVIEs) are much more involved and they deserve to be addressed in another paper. We will report further results along that line in a forthcoming paper.