An optimal control problem of forward-backward stochastic Volterra integral equations with state constraints

This paper is devoted to the stochastic optimal control problems for systems governed by forward-backward stochastic Volterra integral equations (FBSVIEs, for short) with state constraints. Using Ekeland's variational principle, we obtain one kind of variational inequality. Then, by dual method, we derive a stochastic maximum principle which gives the necessary conditions for the optimal controls.


Introductin
As we known, with the exception of the applications in biology, physical, etc, Volterra integral equations often appear in some mathematical economic problems, for example, the relationships between capital and investment which include memory effects (in [24], the present stock of capital depends on the history of investment strategies over a period of time). And the simplest way to describe such memory effects is through Volterra integral operators. Based on the importance of Volterra integral equations, we will study an stochastic optimal control problem about a class of nonlinear stochastic equations−forwardbackward stochastic Volterra integral equations (FBSVIEs, for short). First we review the backgrounds of these two kinds of Volterra integral equations: forward stochastic Volterra integral equations (FSVIEs, for short) and backward stochastic Volterra integral equations (BSVIEs, for short).
Let B(·) be a standard d-dimensional Brownian motion defined on a complete filtered probability space (Ω, F , F, P ), where F = {F t } t≥0 is its natural filtration generated by B(·) and augmented by all the P -null sets in F . Consider the following FSVIE: The readers may refer to [2,7,25,29,30] and the reference cited therein, for the general results on FSVIEs. When studying the stochastic optimal control problems for FSVIEs, we need one kind of adjoint equation in order to derive a stochastic maximum principle. This new adjoint equation is actually a linear BSVIE. This motivates the investigation of the theory and applications of BSVIEs.
As stated in Yong [37], ψ(t) in BSVIE (1) could represent the total (nominal) wealth of certain portfolio which might be a combination of certain contingent claims (for example, European style, which is mature at time T , are usually only F T -measurable), some current cash flows, positions of stocks, mutual funds, and bonds, and so on, at time t. So, in general, the position process ψ(·) is not necessarily Fadapted, but a stochastic process merely F T -measurable. And Yong gave an example to make this point more clear in [37]. Focusing on this kind of position process ψ(·), a class of convex/coherent dynamic risk measures was introduced by Yong in [37] to measure the risk dynamically. Hence, one kind of control problem appears: how to minimize the risk, or how to maximize the utility. Wang, Shi [35] obtained a maximum principle for FBSVIEs without state constraints. In this paper, we study one kind of optimal control problem in which the state equations are governed by the following FBSVIEs: By choosing admissible controls (u, ψ), we shall maximize the following objective functional Our formulation has the following new features: (i) A strong assumption that g(t, ·, ·, ·, ·, ·) in (2) is F t -measurable is given in [35]. By applying the duality principle introduced in Yong [37], we overcome this restriction and assume a natural condition that g(·, s, ·, ·, ·, ·) is F s -measurable.
(ii) ψ in (2) is the terminal state of the BSVIE. In our formulation ψ is also regarded as a control and our control is a pair (u, ψ). In mathematical finance, such kind of controls often appears as "consumptioninvestment plan" (see [32]). For the recent progress of studying this kind of control we refer the reader to [12,16,21,23]. We also impose constraints on the state process Y (·) and ψ.
(iii) We consider the double integral in the cost functional (3) in theory. Some further studies on the applications are still under consideration.
In order to solve this optimal control problem, we adopt the terminal perturbation method, which was introduced in [5,12,[15][16][17][18][19][20][21][22]. Recently, the dual approach is applied to utility optimization problem with volatility ambiguity (see [13,14]). The basic idea is to perturb the terminal state ψ and u directly. By applying Ekeland's variational principle to tackle the state constraints, we derive a stochastic maximum principle which characterizes the optimal control. It is worth to point out that in place of Itô's formula, we need two duality principles established by Yong in [37,38] to obtain the above results. This paper is organized as follows. First, we recall some elements of the theory of BSVIEs in Section 2. In Section 3, we formulate the stochastic optimization problem and prove a stochastic maximum principle. In Section 4, we give two examples. The first example is associated with the model we studied. The last example is about the 'terminal' control ψ(·),

Preliminaries
Let B(·) be a d-dimensional Brownian motion defined on a complete filtered probability space (Ω, F , F, P ), where F = {F t } t≥0 is natural filtration generated by B(·) and augmented by all the P -null sets in F , i.e., where N P is the set of all P -null sets.

Notations
Here we keep on the definitions and notations for the spaces introduced in Yong [38]. For Let S ∈ [0, T ], define the following spaces:

Backward Stochastic Volterra Integral Equations
For the reader's convenience, we present some results of BSVIEs which we will use later. Consider the following integral equation where ψ(·) ∈ L 2 FT (0, T ). We assume: |g(t, s, y, ζ) − g(t, s,ȳ,ζ)| ≤ L(t, s)(|y −ȳ| + |ζ −ζ|), a.s., The following M -solution of BSVIEs was introduced by Yong [38] . (4) holds in the usual Itô sense for almost all t ∈ [S, T ] and, in addition, the following equation holds: For the proof of the following wellposedness results, the readers are referred to Yong [38]. (4) with g and ψ(·) replaced byḡ and ψ(·), respectively, then Yong proved the following two duality principles for linear SVIE and linear BSVIE in [37,38] respectively. And they play a key role in deriving the maximum principle.
be the solution of the following FSVIE: (Y (·), Z(·, ·)) ∈ H 2 [0, T ] be the adapted M -solution to the following BSVIE: Then the following relation holds: is the solution of the following linear BSVIE: and X(·) is the solution of the following FSVIE: Then the following relation holds: For the proofs of Lemmas 3 and 4, the readers are referred to Theorem 5.1 in [38] and Theorem 3.1 in [37], respectively.
, from the wellposedness of BSVIEs (Lemma 2) as well as the proof of Lemma 5, we know that F ε (·, ·) is a continuous function on U.

Examples
First we will give an example associated with the model studied above.
At last we give an example to show the form of the optimal terminal ψ(·).