On stochastic maximum principle for risk-sensitive of fully coupled forward-backward stochastic control of mean-field type with application

In this paper, we are concerned with an optimal control problem where the system is driven by fully coupled forward-backward stochastic differential equation of mean-field type with risk-sensitive performance functional. We study the risk-neutral model for which an optimal solution exists as a preliminary step. This is an extension of the initial stochastic control problem in this type of risk-sensitive performance problem, where an admissible set of controls are convex. We establish necessary as well as sufficient optimality conditions for the risk-sensitive performance functional control problem. Finally, we illustrate our main result of this paper by giving two examples of risk-sensitive control problem under linear stochastic dynamics with exponential quadratic cost function, the second example will be a mean-variance portfolio with a recursive utility functional optimization problem involving optimal control. The explicit expression of the optimal portfolio selection strategy is obtained in the state feedback.


Introduction.
1.1. Motivation example. Modeling and controlling cash flow processes of a firm or a project, such as pricing and managing an insurance contract, is a class of problems where forward-backward stochastic differential equations (FBSDEs in short) provide a natural setup and a powerful tool. In this paper, we shall investigate an example of such a situation arising in the pricing of a simple insurance contract. This example is taken from [2].
A policyholder at an insurance company has paid premiums that at time zero have accumulated to the sum a. The money is invested in an asset portfolio with wealth (x t ) t∈[0,T ] managed by the insurance company under a time interval [0, T ]. At each instant t ∈ [0, T ], the policyholder ought to receive an amount ρ t x t . The present value (price) of the cash stream (ρ s x s ), discounted to time t with a discount factor (deflator) exp − t 0 ς s ds , where ς t is assumed nonnegative, bounded, and deterministic, is given by Assume that the portfolio is invested in a simple Black-Scholes market model consisting of a risk-free asset (for example, a bond or a bank account) with a short interest rate ρ t assumed bounded and deterministic, and a risky asset evolving as a geometric Brownian motion with rate of return χ t and volatility σ t , both assumed to be bounded and deterministic functions of time, with σ t ≥ ε > 0 for all t ∈ [0, T ]. In this market the wealth process (x t ) t∈[0,T ] is governed by the dynamics given by where v t is the amount invested in the risky asset and r t = χ t − ρ t is the risk premium held for this investment. The insurance company allocates the amounts (v t ) in order to come close to the following target at time T : Find the admissible strategies (ρ, v) which maximize the policyholder's preferences represented by the utility function F of the cash streams, under the condition that the total amount to be paid out is equal to the total premium a: (By selecting an appropriate portfolio choice strategy v (.) , where the exponent θ > 0 is called the risk-sensitive parameter. Assume that the policyholder's utility function is of HARA (hyperbolic absolute risk-aversion) type. That is, F (X) = X θ θ , where θ ∈ (0, 1).) We need the following definition of admissible strategies suitable for our problem. An admissible strategy is a pair of (F t ) t≥0 -adapted processes (ρ, v) such that (2) has a strong solution (x t ) t∈[0,T ] that satisfies Now, for each admissible strategy (ρ, u), the (F t ) t≥0 -adapted value process (y t ) t≥0 in (1) satisfies the following BSDE: where (z t ) t≥0 is (F t ) t≥0 -adapted and square-integrable with respect to dt × dP over [0, T ] × Ω. Hence, (2) and (4) satisfied by (x, y, z) is a FBSDE.
The purpose of this paper is to study fully coupled mean-field forward-backward stochastic differential equation with risk-sensitive performance, where the set of an admissible control is assumed to be convex, this is a good extension for the system with cash flow, but we need to add the mean-field processes in all the coefficients of FBSDE, this is not a simple or trivial extension, especially with risk-sensitive performance.
To this end we have to give a brief history of what we want to do?
1.2. Brief historic of mean-field problem. We consider stochastic control problems for state processes governed by a fully coupled FBSDE of mean-field type with risk-sensitive performance functional, which is also called McKean-Vlasov type equation, in the sense the coefficients of the fully coupled mean-field FBSDE are allowed to depend on the state of the process as well as on its expected value. More precisely, the system is defined as for some functions b, σ, and f , and W is a Brownian motion. For every t, the control u t is allowed to take values in some control state space U .
The mean-field SDE is obtained as the mean-square limit, when n → ∞, of a system of interacting particles of the form where, (W i , i ≥ 1) is a collection of independent Brownian motions (see e.g. Sznitman [21] and the references therein). Concerning mean-field backward stochastic differential equations (mean-field BS-DEs), have been first studied by Buckdahn et al, the interested reader is referred to [5,4], the paper Carmona and Delarue [6] was to provide an existence result for the solution of fully coupled FBSDE of the mean-field type. Mathematical mean-field approaches play a crucial role in diverse areas, such as physics, chemistry, economics, finance and games theory, see for example Lasry-Lions [14]. Many papers have been studied the problem of mean-field and established stochastic maximum principle, we can cited here some of them, the first work who gave the necessary optimality conditions was Bukdahn et al [3], after this work many authors have generalized this problem into the others fields of applications, as the paper of Anderson et al [1] they have studied the problem of mean-field type of SDE under the assumptions of a convex action space. Beside, the problem of mean-field has been derived also via Malliavin calculus, the authors Meyer-Brandis, Oksendal and Zhou in paper [17] have been obtained the stochastic maximum principle of mean-field, also to the problem of singular mean-field with a good application to finance we can have the paper of Hu et al [13]. Li in the paper [15], she has been investigated a large extension which is different from the classical ones to mean-field system with an application to linear quadratic problem.
Our control problem consists in minimizing a cost functional with initial and terminal risk-sensitive performance functional, as follows for given functions Φ, Ψ and l. This cost functional is also of mean-field type and with exponential expected, as the functions Φ, Ψ and l depend on the marginal law of the state process through its expected value. The pioneering works on the stochastic maximum principle for this kind of problem was first written by Djehiche et al [10], many works have been studied and continue this problem of risk-sensitive, we can mentioned as an example the papers of Chala [7,8,12].
In the risk-neutral model, the system is governed by the mean-field FBSDE The expected cost of mean-field type to be minimized, the risk-neutral model i.e that without the risk parameter and without the exponential expected A control u is called optimal if Existence of an optimal solution for this problem has been solved to achieve the objective of this paper, and establish necessary and sufficient optimality conditions for these two models, see theorem 3.1 below, we proceed as follows. First, we give -without proof-the optimality conditions for risk-neutral controls. The idea is to use the fact a certain auxiliary state process (m v t ) t∈[0,T ] which is a solution of some SDE, see section 3 below, and transfer the system with two equations the first one is a SDE, whereas the second is a backward SDE, into a system governed by three stochastic differential equations, and the set of risk neutral controls is convex. Then, the adjoint equation with respect to these three equations is given, the proof is a combination of the works of Min et al [18] and those of Yong and Zhou [23,24], the transformation of the adjoint equations will be used as the best approach to solve the risk sensitive control problem, we suggest this transformation to omit the first adjoint equation (which extended from the first SDE of the process) (m v t ) t∈[0,T ] . The necessary and the sufficient optimality conditions have been established with respect only to the second and the third adjoint equations, by the logarithm transformation method, see El Karoui & Hamedene [11], and by using the fact that the coefficients b, σ, f , l, Φ and Ψ are Lipshitzien with respect to their components, the necessary optimality conditions are obtained directly in the global form. A stochastic maximum principle (SMP) for risk-sensitive optimal control problems for Markov diffusion processes with an exponential of integral performance functional was obtained in [16] by making the relationship between the SMP and the Dynamic Programming Principle, the authors have used the first order adjoint process as the gradient of the value function of the control problem. This relationship holds only when the value function is smooth (see Assumption B4 in [16]). By using the smoothness assumption the two papers of [19,20], have extended the approach described above to jump diffusions.
The paper is organized as follows: in Section 2, we formulate precisely our problem, introduce the risk-sensitive model, and state the various assumptions needed throughout this paper. In Section 3, we study our system of fully coupled FB-SDE, the new approach method transformation of the adjoint process is given and studied, stochastic maximum principle for risk-neutral is given. In Section 4, we establish the necessary optimality conditions for risk-sensitive control problem under an additional hypothesis. In Section 5, the sufficient optimality conditions for risk-sensitive performance cost is obtained under the convexity of the Hamiltonian function. In Section 6, we illustrate our main results by two examples; the first is a risk-sensitive control problem under linear stochastic dynamics with exponential quadratic cost function, the second is a financial model of mean-variance with risk-sensitive performance functionnal. Section 7 concludes the paper.

Problem formulation and assumptions.
Let Ω, F, (F t ) t≥0 , P be a probability space filtered satisfying the usual conditions, in which a one−dimensional standard Brownian motion W = (W t ; 0 ≤ t ≤ T ) is given. We assume that F t is de- We denote similarly by S 2 ([0, T ] ; R) the set of continuous one−dimensional random process which satisfy: Let T be a strictly positive real number and U is a nonempty subset of R, such that U is convex.
We denote by U the set of all admissible controls.
For any v ∈ U, we consider the following fully coupled forward-backward stochastic differential equation of mean-field type control system We defined the criterion to be minimized, with initial and terminal risk-sensitive performance functional cost, as follows We use the Euclidean norm |.| in R, is a transpose and T r is trace of matrix. All the equalities and inequalities, mentioned in this paper, are in the sense of dt × dP almost surely on [0, T ] × Ω.
Notation. We use the following notations We assume the following assumptions We also need the following monotonic conditions introduced by [18], are the main assumptions in this papers.
Theorem 2.2. For any given admissible control v (.), and under the above Assumptions 2.1 − 2.3. Then the fully coupled FBSDE of mean-field type control (5) has a unique solution Proof. See [18] Theorem 6 page 3.
A control that solves the problem {(5) , (6) , (7)} is called optimal. Our goal is to establish risk-sensitive necessary as well as sufficient optimality conditions, satisfied by a given optimal control, in the form of mean-field stochastic maximum principle with a risk-sensitive performance functional type.
We also assume that ii) All the derivatives of b, σ, f and l are bounded by iii) The derivative of Φ and Ψ is bounded by Under the above assumptions, for every v ∈ U equation (5) has a unique strong solution, and the cost function J θ is well defined from U into R. For more details of this kind of problem the reader can see paper of Min, Peng and Qin [18].
3. Relation between the risk-neutral and risk-sensitive stochastic maximum principle. The proof of our risk-sensitive stochastic maximum principle necessitates a certain an auxiliary state process m v t which is a solution of the following stochastic differential equation of mean-field type control (SDE of mean-field type control), where Our control problem of {(5) , (6) , (7)} and from the above auxiliary process, the fully coupled forward-backward of mean-field type control is equivalent to We require the following condition and we put also the risk-sensitive loss of functional is given by When the risk-sensitive index θ is small, by Chala et al. [9] the loss functional H (θ, v) can be expanded as where, V ar (Θ T ) denotes the variance of Θ T . If θ < 0, the variance of Θ T , as a measure of risk, improves the performance H (θ, v), in which case the optimizer is called risk seeker. But, when θ > 0, the variance of Θ T worsens the performance H (θ, v), in which case the optimizer is called risk averse. The risk-neutral loss functional E (Θ T ) can be seen as a limit of risk-sensitive functional H (θ, v) when θ → 0, for more details the reader can see the papers [10,11].
In what next let us introduce the following notations.
Notation. For convenience, we will use the following notations throughout this paper, for φ ∈ {b, σ, f, l}, respectively, we define We assume that the Assumptions 2.1 − 2.4 hold, we may apply the SMP for risk-neutral of fully coupled forward-backward of mean-field type control from Min, Peng and Qin [18], and to augmented state dynamics (m u , x u , y u , z u ) to derive the adjoint equation. There exist a unique F t −adapted three pairs of processes (p 1 , q 1 ) , (p 2 , q 2 ) and (p 3 , q 3 ) , which solve the following system matrix of backward SDEs with .
We suppose here that H θ be the Hamiltonian associated with the optimal state dynamics (m u , x u , y u , z u ) , and the pair of adjoint processes ( − → p (t) , − → q (t)) is given by Theorem 3.1. Assume that 2.1−2.4 hold. If (m u , x u , y u , z u ) is an optimal solution of the risk-neutral control problem (8) , then there is three pairs of F t −adapted processes (p 1 , q 1 ) , (p 2 , q 2 ) and (p 3 , q 3 ) that satisfy (9) , such that for all u ∈ U, almost every t and P−almost surely, where H θ v (t) is defined in Notation above.
Proof. We can combine between the proof in the papers [18] and [23,24]. 4. New adjoint equations and necessary optimality conditions. As we said, the Theorem 3.1 is a good SMP for the risk-neutral of fully coupled forwardbackward of mean-field type control problem. We follow the new approach which has been used in [7,8,10,11], and suggest a transformation of the adjoint processes (p 1 , q 1 ) , (p 2 , q 2 ) and (p 3 , q 3 ) in such a way to omit the first component (p 1 , q 1 ) in (9) , and express the SMP in terms of only the last two adjoint processes, that we denote them by ( p 2 , q 2 ) and ( p 3 , q 3 ).
Noting that dp 1 (t) = q 1 (t) dW t and p 1 (T ) = θA θ T , the explicit solution of this backward SDE is As a good view of (12) , it would be natural to choose a transformation of ( − → p , − → q ) into an adjoint process ( p, q) , where p 1 (t) = 1 θV θ t p 1 (t) = 1.
We consider the following transform By using (9) and (13) , we have The following properties of the generic martingale V θ are essential in order to investigate the properties of these new processes ( p (t) , q (t)) .
In the next, we will state and prove the necessary optimality conditions for the system driven by fully coupled FBSDE of mean-field type control with a risksensitive performance functional kind.
As mentioned in the paper of El-Karoui et al. [11], the process Λ θ is the first component of the F t −adapted pair of processes Λ θ , N which is the unique solution to the following quadratic backward SDE of mean-field type To this end, let us summarize and prove some lemmas that we will use thereafter.
where, C T is a positive constant that depends only on T and the boundedness of l, Φ and Ψ.
In particular, V θ solves the following linear backward SDE Hence, the process defined on Ω, F, where is a uniformly bounded F t −martingale.
Proof. Starting with (14). By Assumption 2.4, the functions l, Φ and Ψ are bounded by constant C > 0, we get Therefore, V θ is a uniformly bounded F t −martingale satisfying The solution p, q, V θ , N of the system (18) is unique, such that where Proof. We wish to identify the processes α and β such that By applying Itô's formula to the processes − → p (t) = θV θ t p (t) , and using the expression of V θ in (15) , we obtain By identifying the coefficients the above equation to (21) , and using the relation , the diffusion coefficient β (t) it will be written as and the drift coefficient of the process p (t) Finally, we obtain It is easily verified that In view of (16) , we may use Girsanov's Theorem (see [9], Theorem 2.1 page 115), to claim that In view of (16) and (17) , the probability measures P θ and P are in fact equivalent.
Now replacing (22) in the (23) , to obtain where Therefore, the second and third components of p 2 (t) and p 3 (t) given by (24) , we get The main risk-sensitive second and third adjoint equations for ( p 2 , q 2 ) , ( p 3 , q 3 ) and V θ , N become 5. Sufficient optimality conditions for risk-sen sitive performance cost.
In this section, we study when the necessary optimality conditions (11) become sufficient. For any v ∈ U, we denote by (x v , y v , z v ) the solution of equation (5) controlled by v to state the following result.
convex, and for any v ∈ U such that E Proof. Let u be an arbitrary element of U (candidate to be optimal). For any v ∈ U, we have By applying the Taylor's expansion, and since Φ and Ψ are convex, we get It follows from (9), we remark that p 1 (T ) = θA θ T , , and and Then, by using above inequality in (31) , we obtain In virtue of the necessary optimality conditions (11) , then the last inequality implies that J θ (v) − J θ (u) ≥ 0. Then the theorem is proved.
Remark 5.1. In the last step of proof, and according to (25), we have we know that θV θ t > 0, then the above equation can be rewritten as In virtue of the necessary optimality conditions (26) , then the last inequality implies that J θ (v) − J θ (u) ≥ 0.
6.1. Example 1: Risk-sensitive control applied to the mean-field linearquadratic. We provide a concrete example of the mean field risk sensitive forwardbackward stochastic LQ problem, and we give the explicit optimal control and validate our major theoretical results in Theorem 5.1 (Sufficient optimality conditions for risk-sensitive). First, let the control domain be U = [−1, 1]. Consider the following to the mean-field linear quadratic risk-sensitive control problem subject to where A 1 , A 2 , A 3 , B 1 , B 2 , B 3 , C 1 , C 2 , C 3 , C 4 , C 5 , C 6 and C 7 are constants.
Let (x v t , y v t , z v t ) be a solution of (32) associated with v t . Then, there exist unique F t −adapted three pairs of processes (p 1 , q 1 ) , (p 2 , q 2 ) and (p 3 , q 3 ) of the following FBSDE of mean-field type system (called adjoint equations), according to the equations (9). Now we can define these equations as where We give the Hamiltonian H θ defined by Maximizing the Hamiltonian yields (34) We need only to prove that u t of (34) is an optimal control of (32) . q satisfy (33), then u t is the optimal control of the above mean-field forward-backward stochastic differential equation of linear quadratic problem (32). and By replacing the three above formulas into (35), then we get Then, because of (v t − u t ) being nonnegative, we have the following result Then, (u t p 1 (t) + A 3 p 2 (t) + B 3 q 2 (t) − C 7 p 3 (t)) (v t − u t ) dt .
(36) By replacing u t with its value in (36), we obtain Thus, we get J θ (v t ) ≥ J θ (u t ) , i.e. u t is optimal. This proof is finished.

7.
Conclusion and outlook. This paper contains two main results. The first result (Theorem 4.2), establishes the necessary optimality condition for a system governed by fully coupled FBSDE of mean-field type with risk-sensitive performance, using a scheme almost similar to the one in Chala [7], and Djehiche et al [10]. The second main result, Theorem 5.1, provides sufficient optimality conditions for fully coupled FBSDE of mean-field type with risk-sensitive performance. The proof is based on the convexity conditions of the Hamiltonian function, the initial and terminal terms of the performance function. Note that the risk-sensitive control problems treated by Lim and Zhou [16] are different from ours. We may take as the existing paper which has been established by Djechiche et al [10], and have been generalized into the fully coupled stochastic differential equation with meanfield type, which is motivated by an optimal portfolio choice problem in financial market specially the model of control mean-variance. A problem to be thoroughly addressed in the future, where the system is governed by fully coupled stochastic differential equation of mean-field type with jump diffusion, will be compared with [20]. The maximum principle of risk-neutral obtained by Min et al [18], is similar to ours (Theorem 3.1), but the adjoint equations and maximum conditions depend heavily on the risk-sensitive parameter. If we put θ = λ = µ = κ = 0, we can compare our feedback control of (66) with the control obtained by Hafayed et al [12].