A maximum principle for controlled time-symmetric forward-backward doubly stochastic differential equation with initial-terminal sate constraints

In this paper, we study the optimal control problem of a controlled time-symmetric forward-backward doubly stochastic differential equation with initial-terminal sate constraints. Applying the terminal perturbation method and Ekeland's variation principle, a necessary condition of the stochastic optimal control, i.e., stochastic maximum principle is derived. Applications to backward doubly stochastic linear-quadratic control models are investigated.

In order to produce a probabilistic representation of certain quasilinear stochastic partial differential equations (SPDEs), Pardoux and Peng [17] first introduced a class of stochastic differential equations, i.e. backward doubly stochastic differential equations (BDSDEs for short) and proved the existence and uniqueness theorem of BDSDEs. Using such BDSDEs they proved the existence and uniqueness theorem of those quasilinear SPDEs and thus significantly extended the famous Feynman-Kac formula for such SPDEs.
Connecting the theory of FBSDEs and BDSDEs, Peng and Shi [18] studied the following time-symmetric forward-backward stochastic differential equations: x t , z t , y t , q t , u t )dt + G(t, x t , z t , y t , q t , u t )dW t − z t dB t , 0 ≤ t ≤ T, x t , z t , y t , q t , u t )dt + g(t, x t , z t , y t , q t , u t )dB t − q t dW t , 0 ≤ t ≤ T, Here the forward equation is "forward" with respect to a standard stochastic integral dW t , as well as "backward" with respect to a backward stochastic integral dB t ; the coupled "backward equation" is "forward" under the backward stochastic integral dB t and "backward" under the forward one, which generalized the general FBSDEs. In other wards, both the forward equation and the backward one are BDSDEs with different directions of stochastic integral. Under certain monotonicity conditions, they proved the uniqueness and existence theorem for these equations.
It is well known that the maximum principle is an important approach to study optimal control problems. The systematic account on this theory can be found in [2] and [22]. When the controlled system under consideration is assumed to be with state constraints, especially with sample-wise constraints, the corresponding stochastic optimal control problems are difficult to solve. A sample-wise constraint requires that the state be in a given set with probability 1; for example, a nonnegativity constraint on the wealth process, i.e., bankruptcy prohibition in financial markets. In order to deal with such optimal control problems, an approach named "terminal perturbation method" was introduced and applied in financial optimization problems recently (see [9][10][11][12]). This method is based on the dual method or martingale method introduced by Bieleckiet in [3] and El Karoui, Peng and Quenez in [6]. It mainly applies Ekeland's variational principle to tackle the state constraints and derive a stochastic maximum principle which characterizes the optimal solution. For other works about the optimization problem with state constraints, the readers may refer to [20][21]. In this paper, a stochastic maximum principle is obtained for the controlled time-symmetric FBSDEs with initial-terminal state constraints by using Ekeland's variational principle. Different from [7], our controlled system is time-symmetric forward-backward stochastic differential equations and sample-wise initial and terminal state constraints are considered.
We give three specific cases to illustrate the applications of our obtained results. In the first case, the controlled system equations are composed of a normal Forward SDE and a Backward doubly SDE. We only consider one Backward doubly SDE as our system equation in the second case. Finally we study the backward doubly stochastic linear-quadratic (LQ) problems. This paper is organized as follows. In section 2, applying Ekeland's variation principle we obtain a stochastic maximum principle of this controlled time-symmetric forward-backward stochastic differential equations with initial-terminal state constraints. Some applications are given in the last section.

The main problem § 2.1 Preliminaries
Let us first recall the existence and uniqueness results of the backward doubly stochastic differential equations (BDSDEs) which was introduced by Pardoux and Peng [17], and an extension of the well known Itô's formula which would be often used in this paper.
Let (Ω, F , P) be a probability space, and T > 0 be fixed throughout this paper. Let {W t , 0 ≤ t ≤ T } and {B t , 0 ≤ t ≤ T } be two mutually independent standard Brownian motion processes, with values respectively in R d and in R l , defined on (Ω, F , P). Let N denote the class of P-null set of F . For each t ∈ [0, T ], we define: T ]} is neither increasing nor decreasing, and it does not constitute a filtration.
For any Euclidean space H, we denote by < ·, · > the scale product of H. The Euclidean norm of a vector y ∈ R k will be denoted by |y|, and for a d × n matrix A, we define ||A|| = T r(AA * ).
For any n ∈ N, let M 2 (0, T ; R n ) denote the set of (classes of dP ×dt) a.e. equal) n-dimensional jointly measurable random processes {ϕ t ; t ∈ [0, T ]} which satisfy: We denote by S 2 (0, T ; R n ) the set of continuous n-dimensional random processes which satisfy: be jointly measurable and such that for any (y, q) ∈ R k × R k×d , f (·, y, q) ∈ M 2 (0, T ; R k ), g(·, y, q) ∈ M 2 (0, T ; R k×l ).
Moreover, we assume that there exist constants C > 0 and 0 < α < 1 such that for any (ω, Given η ∈ L 2 (Ω, F T , P;R k ), we consider the following Backward doubly stochastic differential equation: We note that the integral with respect to {B t } is a "backward Itô integral" and the integral with respect to {W t } is a standard forward Itô integral. These two types of integrals are particular cases of the Itô-Skorohod integral, see Nualart and Pardoux [15].
Next let us recall an extension of the well known Itô's formula in [17] which would be often used in this paper. Then,

(2.3)
Our optimization problem is: We shall denote by N(a, b) the set of all feasible (ξ, η, u(·)) for any given a and b.
The aim of this paper is to obtain a characterization of (ξ * , η * , u * (·)), i.e. a stochastic maximum principle. § 2.3 Stochastic Maximum Principle Using Ekeland's variational principle, we derive maximum principle for the optimization problem (2.4) in this section. For simplicity, we first study the case where l(y(t), z(t), y(t), q(t), u(t), t) = 0, χ(x) = 0 and λ(y) = 0 in subsection 2.3.1-2.3.3, and then present the results for the general case in subsection 2.3.4.
To derive the first-order necessary condition, we let (x(·),ẑ (·),ŷ(·),q(·)) be the solution of the following time-symmetric forward-backward differential equations: We have the following convergence.

Variational inequality
In this subsection, we apply Ekeland's variational principle [4] to deal with initial-terminal state constraints E(ψ(x 9) where a and b are the given initial and terminal state constraints and ε is an arbitrary positive constant.
It is easy to check that the mapping |E(ψ(x ) and γ(y (ξ,η,u(·)) 0 ) are all continuous functionals from U to R.
From (iii), we know that (2.11) On the other hand, similarly to lemma 2.4 we have This leads to the following expansions . Applying the linearization technique, then So we have the following expansions For the given ε, we consider the following four cases: Case 1. There exists ρ 0 > 0 such that In this case, Dividing (2.11) by ρ and sending ρ to 0, we obtain √ εd((ξ ε , η ε , u ε (·)), (ξ, η, u(·))).
Since the proof of the maximum principle is essentially similar as in the preceding subsection, we only present the result without proof.
Remark: Let us denoted the boundary of K 1 by ∂K 1 . Set Then Similar analysis can be used to the boundaries of K 2 and K.

Applications
In this section, we give three specific cases to illustrate the applications of our obtained results. § 3.1 Systems composed of a Forward SDE and a BDSDE Classical formulation For given ξ ∈ L 2 (Ω, F 0 , P;R n ) and u(·) ∈ U[0, T ], we consider the following controlled system composed of a Forward SDE and a Backward doubly SDE.
where b ∈ R k is given, x 0 = ξ ∈ K 1 , a.s where K 1 is a given nonempty convex subset in R n .
Note (H1) and (H4) imply the mapping is a bijection from R d×n on to itself for any (y, t).
A key observation that inspires our approach of solving problem (3.2) is that, since u → σ(x, u, t) is a bijection, q(·) can be regarded as the control; moreover, by the BSDE theory selecting q(·) is equivalent to selecting the terminal value y T . Hence we introduce the following "controlled" system: where the control variables are the random variables ξ and η to be chosen from the following set For each (ξ, η) ∈ U, consider the following cost where l(x, z, y, q, t) =l(x, z, y ,σ (y, q, t), t).
This gives rise to the following auxiliary optimization problem: is the solution of (3.3) at time 0 under ξ and η. It is clear that the original problem (3.2) is equivalent to the auxiliary one (3.4).
Hence, hereafter we focus ourselves on solving (3.4). The advantage of doing this is that, since ξ and η now are the control variable, the state constraint in (3.2) becomes a control constraint in (3.4), whereas it is well known in control theory that a control constraint is much easier to deal with than a state constraint. There is, nonetheless, a cost of doing so is that the original initial condition y (ξ,η) 0 = b now becomes a constraint, as shown in (3.4).
Then from (2.14), the adjoint equations become