Optimal Control Problems of Forward-Backward Stochastic Volterra Integral Equations

Optimal control problems of forward-backward stochastic Volterra integral equations (FBSVIEs in short) are formulated and studied. A general duality principle is established for linear backward stochastic integral equation and linear stochastic Fredholm-Volterra integral equation with mean-field. With the help of such a duality principle, together with some other new delicate and subtle skills, Pontryagin type maximum principles are proved for two optimal control problems of FBSVIEs.


Introduction
Let (Ω, F , F, P) be a complete filtered probability space on which a standard one-dimensional Brownian motion {W (t), t ≥ 0} is defined with F = {F t t ≥ 0} being its natural filtration augmented by all the P-null sets. We point out that assuming W (·) to be one-dimensional is just for the simplicity of presentation; Our results remain for the case of multi-dimensional Brownian motions.
Let us start with a classical stochastic optimal control problem. To this end, we consider the following controlled stochastic differential equation (SDE, for short): (1.1)    dX(s) = b(s, X(s), u(s))ds + σ(s, X(s), u(s))dW (s), s ∈ [0, T ], with cost functional (1.2) J 0 (x; u(·)) = E h(X(T )) + T 0 g(s, X(s), u(s))ds , where b, σ, h, g are some suitable maps, the control u(·) is taken from some suitable set U, and the state process X(·) is valued in R n . In the above, components of X(·) could be wealth, owned commodities/assets, 1 inventory of products, and some economic factors (interest rates, unemployment rate), and so on. Also, u(·) can be regarded as some kind of investment, production effort, labor force, transaction of assets, etc. Then our classical optimal control can be stated as follows.
Standard results for the above Problem (C) 0 can be found, say, in [19]. Now let us make some further analysis on the equation (1.1) which can be written as follows: (1.4) Although the state equation has the above integral form, it is still memoryless, in the sense that the increment X(t+∆t)−X(t) of the state on [t, t+∆t] only depends on the local "driving force" b(s, X(s), u(s)), σ(s, X(s), u(s)) , s ∈ [t, t + ∆t] : X(t + ∆t) − X(t) = t+∆t t b(s, X(s), u(s))ds + t+∆t t σ(s, X(s), u(s))dW (s).
Whereas, in reality, memory or long-term dependence often exists. In another word, the increment X(t + ∆t) − X(t) of the state on [t, t + ∆t] might depend on the "driving force" of non-infinitesimal time duration, say, [t − τ, t], for some τ > 0. For example, the current production level usually depends on some renovation of production equipments some time ago, the profit of investment usually depends on the transactions some time before, the air pollution is caused by some bad production strategies some years ago, etc. To model various possible situations with memory, instead of (1.4), we may consider the following controlled (forward) stochastic Volterra integral equation (FSVIE, for short): (1.5) X(t) = ϕ(t) + Unlike (1.4), due to the dependence of b(t, s, x, u), σ(t, s, x, u) on t, even for the case ϕ(t) ≡ x, we have X(t + ∆t) − X(t) = t+∆t t b(t + ∆t, s, X(s), u(s))ds + t+∆t t σ(t + ∆t, s, X(s), u(s)dW (s) + t 0 b(t + ∆t, s, X(s), u(s)) − b(t, s, X(s), u(s)) ds + t 0 σ(t + ∆t, s, X(s), u(s)) − σ(t, s, X(s), u(s)) dW (s), which depends not only on the values of the "driving force" in [t, t + ∆t], but also on those in the whole interval [0, t] up to the current time t. Therefore, with suitable choices of b and σ, it is possible to model certain memory effects through (1.5). Based on these arguments, one can use deterministic or stochastic Volterra integral equations to describe some economic models, see [3,8,4], for examples.
On the other hand, let us turn to the cost functional (1.2). As we know, stochastic differential utility (SDU, for short) introduced by Duffie-Epstein ( [2]) can be represented by backward stochastic differential equations (BSDEs, for short). More precisely, if C(·) is a consumption process and ξ is a payoff at the terminal time T , then an SDU process Y (·) for the pair (C(·), ξ) can be modeled by the following: (1. 6) Y (t) = E ξ + T t g(s, Y (s), C(s))ds F t , t ∈ [0, T ], for some suitable map g. This can also be regarded as a dynamic risk measure process associated with the pair (ξ, C(·)). It turns out that (1.6) admits the following equivalent form, which is a BSDE whose solution is a pair (Y (·), Z(·)) of F-adapted processes ( [8,7,19]). Note that if we let ξ = h(X(T )), u(·) = C(·), then the cost functional (1.2) admits the following representation: Thanks to the further development of BSDEs, one could extend the SDU theory via more general BSDEs, namely, one may define an SDU process (Y (·), Z(·)) as the adapted solution to the following general BSDE: Y (T ) = ξ, whose equivalent integral form reads Although it is very general, the above is still challenged by the following two aspects: (i) The terminal payoff/cost ξ is time-independent; (ii) the "running utility/cost" is of memoryless feature. These two lead to the time-consistency of the utility process Y (·), which is a little too ideal. In reality, substantial evidence (see [1], for example) shows that people in the real life are more concerned (or impatient) about the choices (or decisions) for the immediate future, but are more rational (or patient) when facing long-term alternatives. Such a phenomenon is just one particular case of time inconsistency. Therefore inspired by the theory of backward stochastic Volterra integral equations (BSVIEs, for short) ( [6,13,14,15,16,17,18]), we could modify the above into the following form, taking into account of our controlled FSVIE: s, X(t), X(s), Y (s), Z(t, s), Z(s, t), u(s))ds where (Y (·), Z(· , ·)) is a so-called adapted M-solution of the above (see [18]). In the above, we see that both X(t) and X(T ) appear in the free-term ψ(t, X(t), X(T )). A motivation of that is the following: Suppose X(·) represent the production level process of certain product. One expects that the terminal level should be within a certain range determined by the current level, due to the limitation of resource, manpower, machine capacity, market demand/price, etc. Some similar explanations can be made for the appearance of both X(t) and X(s) in the integrand.
Motivated by the above arguments, in this paper we study the following controlled forward-backward stochastic Volterra integral equations (FBSVIEs, for short): Y (t) = ψ(t, X(t), X(T )) + T t g(t, s, X(t), X(s), Y (s), Z(t, s), Z(s, t), u(s))ds We call (X(·), Y (·), Z(· , ·)) the state and u(·) the control. In such a system, X(·) and Y (·) can be regarded as the portfolio process and the dynamic risk process, respectively. To introduce the cost functional, we need to separate two cases.
First of all, if the generator g(·) of the BSVIE in (1.10) is independent of Z(s, t), then the state equation reads: In this case, under some mild conditions, for any control u(·), there exists a unique triplet (X(·), Y (·), Z(· , ·)), called the adapted solution to (1.11), such that and (1.11) is satisfied in the usual Itô's sense. Moreover, Y (·) ∈ C F (0, T ; L 2 (Ω; R n )). Therefore, Y (0) is well-defined and for such a case, we may introduce the following cost functional: Note that in the above case, the process Z(t, s) is only defined for 0 ≤ t ≤ s ≤ T .
On the other hand, if the generator g(·) depends on Z(s, t), then, by [18], under some suitable conditions, there exists a unique triplet (X(·), Y (·), Z(· , ·)), called the adapted M-solution to (1.10), such that (1.14) and in addition to (1.10) being satisfied in the usual Itô's sense, one also has Different from the first case, in this second case, the process Z(t, s) is defined on [0, T ] 2 , and the additional relation (1.15) holds. In this second case, due to the fact that t → Z(s, t) (for t ≤ s) is not necessarily continuous, we could not expect the continuity of t → Y (t). Therefore, Y (0) might not be well-defined in general. Consequently, the corresponding cost functional should not contain the term like h(X(T ), Y (0)) as in J 1 (u(·)). Fortunately, a comparison theorem found in [14] suggests that in the current case, it might be more proper to use E T 0 Y (s)ds as an alternative for Y (0). Hence, we propose the following cost functional: s, X(s), Y (s), Z(t, s), u(s))dsdt .
From the above, we see that one can formulate two different optimal control problems. In this paper, we will establish Pontryagin type maximum principles for the optimal control problems corresponding to 4 the above two settings. It is known that in deriving maximum principle, besides the suitable variation of the state equation and cost functional, the key is to have a duality principle. The major contribution of this paper is the discovery of a duality principle for general linear BSVIEs, which is a significant extension of that presented in [18]. It turns out that our new duality principle involves a special type of stochastic Fredholm-Volterra integral equations, and we are able to obtain its solvability under natural conditions. It is worthy of pointing out that in contract with SDE case ( [9,10,19]), we need to carry out all the calculations without differentiation due to the lack of Itô's formula for stochastic integral equations.
The rest of this paper is organized as follows. In Section 2, some basic results concerning BSVIEs are recalled. In Section 3, we state two maximum principles for controlled FBSVIEs. A general dual principle for linear BSVIEs is established in Section 4. Then the stated maximum principles are proved in Section 5. Section 6 concludes the paper.

Results for BSVIEs Revisited
In this section, we are going to recall some relevant results for BSVIEs. To this end, let us first introduce some spaces. For H = R n , R n×m , etc., we denote its norm by | · |. For 0 ≤ s < t ≤ T , define The spaces L 2 F Ω; C(s, t; H) and C F s, t; L 2 (Ω; H) can be defined in the same way. It should be pointed out that L 2 FT (Ω; C(s, t; H)) ⊆ C FT (s, t; L 2 (Ω; H)), L 2 F (Ω; C(s, t; H)) ⊆ C F (s, t; L 2 (Ω; H)), and the equalities do not hold in general. Further, we denote

5
We denote . Further, we let M 2 [0, T ] be the set of all (y(·), z(· , ·)) ∈ H 2 [0, T ] such that Clearly, M 2 [0, T ] is a closed subspace of H 2 [0, T ]. Also, for any (y(·), z(· , ·)) ∈ M 2 [0, T ], we have The above implies that for any β ≥ 0, there exists a constant K > 0 depending on β such that We now consider the following two types of BSVIE: Note that (2.4) is a special case of (2.3) with the generator g(·) independent of Z(s, t). We recall the following definition. We point out that for BSVIE (2.4) the values Z(t, s) of Z(· , ·) with t ≥ s are irrelevant. Hence, adapted solutions (y(·), z(· , ·)) of (2.4) need only belong to H 2 The following is a hypothesis for the coefficients of BSVIE (2.3).
On the other hand, similar to (2.19), we have which leads to the estimate (2.8).
(ii) Let (Y (·), Z(· , ·)) be the adapted solution of BSVIE (2.4). For any fixed t ∈ [0, T ], we let (η(t, ·), ζ(t, ·)) be the adapted solution to the following BSDE: Then we know that By (2.20), we have Thus, By Gronwall's inequality, we obtain estimate (2.10). This also leads to Similar to (2.17), in the current case, we have Then applying Gronwall's inequality, we obtain stability estimate (2.11). To prove the continuity of t → Y (t), we let t, t ′ ∈ [0, T ] and consider the following: Then the stability of adapted solutions to BSDEs implies that Hence, (t, r) → η(t, r) is continuous, i.e.,

Optimal Control Problems and Maximum Principles
Now, we consider the following controlled FBSVIE: where admissible control u(·) belongs to U[0, T ] defined by with U being a nonempty convex subset of R ℓ . For convenience, we let 0 ∈ U . Also, we will consider the following FBSVIE which is a special case of (3.1): s, X(s), u(s))ds + t 0 σ(t, s, X(s), u(s))dW (s), s, X(t), X(s), Y (s), Z(t, s), u(s))ds is called an adapted M-solution of (3.1) if X(·) satisfies the forward stochastic Volterra integral equation (FSVIE, for short) in (3.1) and (Y (·), Z(·, ·)) is the adapted M-solution of the BSVIE in (3.1). Also, a triple is called an adapted solution of (3.2) if X(·) satisfies the FSVIE in (3.2) and (Y (·), Z(·, ·)) is an adapted solution of BSVIE in (3.2).
The following collects some basic assumptions on the coefficients of FBSVIE (3.1).
First, we consider state equation (3.2). Since for such a case, Y (0) is well defined, we may introduce the cost functional as follows: For the involved functions h and f in (3.9), we impose the following hypothesis.
The notations σ x (t, s), σ u (t, s), ψ x ′ (t), ψ x (t), g x ′ (t, s), g x (t, s), g y (t, s), g z (t, s), g z ′ (t, s), g u (t, s), h x , h y , f x (t, s), f y (t, s), and f u (t, s) are similar. Also, for any scalar valued function, say x → f (t, s, x, y, z, u), f x (t, s, x, y, z, u) is regarded as row vector, i.e., R 1×n -valued. Such a convention will be consistent with vector valued functions, say, x → ψ(t, x, x ′ ) for which ψ x (t, x, x ′ ) takes values in R m×n . We now state the following maximum principle.
Note that in (3.12), λ(·) solves an FSDE, ξ(·) solves a special type of stochastic Fredholm integral equation with mean-field. We will show in the next section that such an equation admits a unique solution ξ(·) which is not required to be F-adapted. The equation for (µ(·), ν(·)) is a BSDE, and that for (p(·), q(· , ·)) is a BSVIE. We see that the system (3.12) is a decoupled system. Next let us consider the state equation (3.1). For this case, we introduce the following cost functional: For the involved functions h and f in (3.13), we impose the following hypothesis.
We may pose the following problem.
where (ξ(·), µ(·), ν(·), p(·), g(· , ·)) solves the following adjoint equation: We see that different from Theorem 3.2, in the above the adjoint equation only consists of three equations, the equation for λ(·) is not necessary here. Actually, we will see that the equation for λ(·) is used to take care of the term involving Y (0) in the cost functional. Again, (3.15) is also decoupled.

Duality Principles
The aim of this section is to establish a duality principle between the following linear BSVIE: and its adjoint equation, where (Y (·), Z(·, ·)) is the unique M-solution associated with ψ(·) ∈ L 2 FT (0, T ; R m ). We introduce the following hypothesis for the coefficients of the above equation. By Theorem 2.3, we know that under (H7), for any ψ(·) ∈ L 2 FT (0, T ; R m ), linear BSVIE (4.1) admits a unique adapted M-solution (Y (·), Z(· , ·)) ∈ M 2 [0, T ]. Note that in [18] (see Theorem 5.1 there), a duality principle was established for the case that Z(t, s) does not appear (or B(· , ·) = 0). The significance here in the current paper is that we have discovered the adjoint equation of (4.1) with all the interested terms appearing and we have the well-posedness of such an equation. We now introduce the adjoint equation for (4.1). For any (α(·), β(· , ·)) ∈ L 2 F (0, T ; R m ) × L 2 (0, T ; L 2 F (0, T ; R m )), consider the following stochastic integral equation: for any integrable random variable ζ and r ∈ [0, T ]. We call (4.2) the adjoint equation of linear BSVIE (4.1). It is seen that (4.2) is a mean-field stochastic Fredholm-Volterra type integral equation with some special structure, whose unknown is an F T -measurable process ξ(·). Unlike usual BSDEs or BSVIEs, in the above, we do not require ξ(·) to be F-adapted. We now state the duality principle.

one obtains
Now, for any µ > 0, it follows that Since K in the above is an absolute constant, by choosing µ > 0 large, we get that the mapξ(·) → ξ(·) is a contraction in L 2 FT (0, T ; R m ) with a weighted norm. Hence, it admits a unique fixed point which is the unique solution of (4.4).
Let us recall the following duality principle found in [18], which is a corollary of Theorem 4.1.

Proofs of Theorem 3.2 and Theorem 3.3
This section is devoted to the proofs of Theorems 3.2 and 3.3.
As a standard step, to obtain the maximum principle we need to obtain the variation of the state and the cost functional with respect to the control and then use duality principle(s).
For Theorem 3.1, we have the following result.
We now carry out a proof of Theorem 3.2.