A MEAN-FIELD STOCHASTIC LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEM WITH JUMPS UNDER PARTIAL INFORMATION∗

In this article, the stochastic linear-quadratic optimal control problem of mean-field type with jumps under partial information is discussed. The state equation which contains affine terms is a SDE with jumps driven by a multidimensional Brownian motion and a Poisson stochastic martingale measure, and the quadratic cost function contains cross terms. In addition, the state and the control as well as their expectations are contained both in the state equation and the cost functional. This is the so-called optimal control problem of mean-field type. Firstly, the existence and uniqueness of the optimal control is proved. Secondly, the adjoint processes of the state equation is introduced, and by using the duality technique, the optimal control is characterized by the stochastic Hamiltonian system. Thirdly, by applying a decoupling technology, we deduce two integro-differential Riccati equations and get the feedback representation of the optimal control under partial information. Fourthly, the existence and uniqueness of the solutions of two Riccati equations are proved. Finally, we discuss a special case, and establish the corresponding feedback representation of the optimal control by means of filtering technique. Mathematics Subject Classification. 93E20, 60H10. Received June 28, 2021. Accepted May 1, 2022.


Introduction
The so-called optimal control is to find the optimal control plan among the possible control schemes, so that the control system can achieve the desired goal optimally.The research on optimal control theory has a long history.As early as the early 1950s, Bushaw studied the time-optimal control of servo systems, and then LaSalle developed the time-optimal control theory, the so-called Bang-Bang control theory was proposed.From 1953 to 1957, American scholar Bellman established the theory of dynamic programming and developed the Hamilton-Jacobi theory of variation.From 1956 to 1958, the former Soviet Union scholar Pontryagin and others established the principle of maximum.As we all know, Pontryagin's maximum principle and Bellman's dynamic programming principle were the two main methods for solving stochastic optimal control problems, and also the most commonly used methods.With the continuous in-depth research of aerospace, navigation, aviation and guidance technology, the optimization of system has become an important issue.Optimal control theory has also made great progress and become a very important branch of modern control theory.
On the basis of optimal control, when the state equation of the control system is a linear equation and the cost functional is quadratic, the optimal control can be given in the form of linear feedback.Such a problem is called the Linear-Quadratic (LQ) optimal control problem.Getting the feedback representation of optimal control is the most basic task.The LQ optimal control problem was first studied by Bellman-Glicksberg-Gross in 1958.In 1960, Kalman established the state feedback optimal control, and introduced the Riccati equation into the control theory.In 2000, [9] studied the relationship between stochastic control problems and the backward stochastic differential equation (BSDE) for the first time.Later, on the basis of [9], the explicit form of optimal control for LQ problem of BSDE was obtained by Lim and Zhou in [11].In 2016, Sun et al. in [17] investigated the open-loop and closed-loop solvability of LQ optimal control problem.
The LQ optimal control problem is widely used and can be applied to many aspects, but when dealing with unexpected situations in financial problems, it is necessary to use a jump system to characterize.Therefore, the discussion on LQ problem with jumps is also very important.In [2], Boel, Varaiya et al. discussed the optimal control problem of the process with jumps for the first time.Tang and Li in [19] studied the essential conditions for optimal control of stochastic systems with random jumps and first discussed the BSDE with Poisson process.In 2003, Wu and Wang studied the stochastic problem of the system driven by Brownian motion and Poisson jump in [20].In 2014, Meng [13] explored the solutions of the backward stochastic Riccati equations exist and were unique.In 2020, Zhang et al. [24] studied the solvability of matrix valued backward stochastic Riccati equations with jumps, which are associated with a stochastic linear quadratic optimal control problem with random coefficients and driven by both Brownian motion and a Poisson processes.In 2021, Moon et al. [14] studied the indefinite linear quadratic stochastic optimal control problem for stochastic differential equations with jump diffusions and random coefficients driven by both the Brownian motion and (compensated) Poisson processes.
Nowadays, in the stochastic control theory, the systems with interactions have attracted more and more attentions, because it actually described a phenomenon in reality: Interaction exists in the whole system, so the state of a single individual depends not only on its own state, but also on the mean-value of the whole system.And the model is designed to study such systems is called mean-field (MF) model, then it began to be widely applied in many fields such as statistical physics, biology, financial engineering, social science, etc.In 1956, Kac [8] considered the MF-SDE in the first time.The control problems with mean-field type have become more and more popular after Buckdahn et al. [3].In 2008, Hu and Oksendal [5] researched the LQ problem with random coefficients under partial information driven by a Poisson jumps, and proved that the optimal control has a state feedback representation by a BSDE with jumps.In 2011, Yong [22] dealt with a LQ optimal control problem of MF-SDE with deterministic coefficients driven by Brownian motion in finite horizon, by using variational method, the optimality system was derived.And by using decoupling technology, he got the optimal control in the feedback form.In 2015, Yong et al. [6] studied an LQ optimal control problem of MF-SDE in infinite horizon.Aimed at MF-LQ optimal control problem, in 2017, Sun [16] explored the relationship between the open-loop solvability and the (uniformly) convexity of the cost functional.In 2019, Meng and Tang [18] explored the optimal control problem of the mean-field type driven by Brownian motion and a Poisson stochastic martingale measure.The existence and uniqueness of optimal control was obtained by the classical convex variational principle [4], two Riccati equations were deduced, and the state feedback representation of optimal control was also obtained.
Besides, when we make some decisions, it is common that the information we observed is not always complete, thus the research of MF-LQ under partial information is getting deeper and deeper.This kind of problem is related to filtering theory.A systematic introduction to the theory and application of linear filtering can be consulted in Bensoussan [1].Xiong in 2008 [21] introduced the stochastic filtering theory systematically.Oksendal and Sulem [15] dealt with the case when the state equation is described by a forward-backward SDE (FBSDE) with random jump under partial information.Ma and Liu [12] researched on LQ optimal control problem of FBSDE with mean-field type under partially observed.For LQ optimal control problem of MF-BSDE, Li et al. [10] concluded that the optimal control can be represented by two Riccati equations and a MF-SDE.In 2020, under partial information, Zhang [24] studied the optimal control problem of MF-SDE with terminal constraint.Huang et al. [7] in 2020 considered LQ optimal control problem of BSDE under partial information with its application, they also extended BSDE to MF-BSDE, and obtained the corresponding feedback form of optimal control under partial information.
In this paper, we discuss a kind of mean-field stochastic linear-quadratic (MF-LQ) optimal control problem with jumps under partial information.Comparing with the classical LQ optimal control problem, the features of this case are as follows: (1) the state equation contains affine terms is a SDE with jump driven by a multidimensional standard Brownian motion and a Poisson stochastic martingale measure; (2) the quadratic cost function contains cross terms; (3) mean-field type, the state equation and the cost functional also conclude the expectations of state X(•) and the control u(•); (4) under partial information, the state X(•) is F t -adapted, and The rest of the article is arranged as follows.In Section 1, we introduce some basic notations used throughout the article, state the problem, and show our needed assumptions.Secondly, through the classical convex variation principle, we prove the existence and uniqueness of the optimal control.In Section 3, we introduce the adjoint processes of the state equation, by the duality method, a stochastic Hamiltonian system is derived to characterize the optimal control.Fourthly, by using decoupling technology, we deduce two stochastic integro-differential Riccati equations and the corresponding state feedback representation of optimal control.In Section 5, we prove the existence and uniqueness of the solutions of two Riccati equations.And we discuss a special case of MF-LQ optimal control problem under partial information, by means of filtering technique, and get the corresponding feedback representation of optimal control in Section 6.

Notations
Throughout this paper, given a fixed T > 0, let (Ω, F , P) be a complete probability space equipped with a right-continuous P-complete filtration F := {F t | t ∈ [0, T ]}, to be specified later.Furthermore, we assume that F T = F .Denote by P the predictable σ-algebra on Ω × [0, T ] associated with F, and by B(Λ) the Borel σ-algebra of any topological space Λ.Let {W (t)} 0≤t≤T = {W 1 (t), W 2 (t), • • • , W d (t)} 0≤t≤T be a given ddimensional standard Brownian motion.Let (Z, B(Z), ν) be a measure space with ν(E) < ∞ for any E ∈ B(Z), and η be a stationary Poisson point process with the characteristic measure ν.Then, the counting measure induced by η is defined by and then μ(de, dt) µ(de, dt) − ν(de)dt is the compensated Poisson random martingale measure which is assumed to be independent of the Brownian motion {W (t)} 0≤t≤T .Moreover, the filtration F is assumed to be the P-augmentation of the natural filtration generated by the Brownian motion {W (t)} 0≤t≤T and the Poisson random measure {µ(( Also, under many situations, the full information F t is inaccessible for control and ones can only observe a partial information.In this case, the admissible control process u(•) is assumed to be a G t -prediction process, here G t ⊆ F t .We set G = {G t } 0≤t≤T is a given subfiltration which represents the information available to the controller at time t.For example, we can set G t = F t−δ , where δ > 0, is a fixed delay of information.
Let's introduce some basic notations needed throughout this paper.
-R n : the n-dimensional Euclidean space.
-R n×m : the space of all n × m real matrices.
-M : the transpose of matrix M .
-N −1 : the inverse of matrix N .
-M, N = tr(M N ): the inner product on R n×m .
-|M | = tr(M M ): the induced norm of M .
-S n ∈ R n×n : the space of all n × n symmetric matrices. - -L 2 F (Ω; L 1 (t, T ; H)): the space all H-valued and F-progressively measurable processes g : -L ν,2 (Z; H): the space of all H-valued measurable function r : Z → H defined on the measure space (Z, B(Z); v) satisfying -L ν,2 F ([t, T ] × Z; H): the space of all L ν,2 (Z; H)-valued and F s -predictable processes r :

Formulation of the problem
Now we discuss a linear system driven by a d-dimensional standard Brownian motion W (t) and a Poisson random martingale measure {μ(dt, dθ)} 0≤t≤T , in addition, the system is mean-field type with partial information.Within the time interval [t, T ], the state equation is described by the following MF-SDE: •) are given deterministic matrixvalued functions; b(•), σ i (•), h(•, •) are vector-valued F t -predictable processes; The mathematical expectation is denoted by E, E[X(s)] and E[u(s)] are called the mean-field terms of equation (1.1).And the solution of (1.1) denoted by X(•) or X x,u (•) valued in R n , is called the state process; u(•) valued in R m is said to be the control process required to be G t -predictable.
For any t ∈ [0, T ], we introduce the following Hilbert space: where u(•) ∈ A is called the admissible control process, and (X(•), u(•)) is called an admissible pair.
Next we consider the quadratic cost functional with cross terms as follows: where G, Ḡ are symmetric and nonnegative; are given deterministic matrix-valued functions; g is an F T -measurable random variable, ḡ is a deterministic vector; q(•), ρ(•) are allowed to be vector-valued F t -adapted processes, and q(•), ρ(•) are deterministic vector-valued functions.Then we can formulate the MF-LQ optimal control problem under partial information.
Proposition 1.1.(Partial Information MF-LQ) Under partial information, for any given t ∈ [0, T ], x ∈ R n , in order to minimize the cost functional, to find an admissible control u * (•) ∈ A such that It is generally accepted for the initial pair (t, x) ∈ [0, T ] × R n , any u(•) ∈ A satisfying the above conditions is called the optimal control of the Problem 1.1, the corresponding solution F (t, T ; H) is called an optimal state process, and the pair (X * (•), u * (•)) is called an optimal pair.The function we denote the corresponding cost functional by J 0 (t, x; u(•)), the corresponding value function by V 0 (t, x).At the moment, the cost functional can be represented as: Comparing with the general state equation (1.1), when the initial pair is (t, 0), in other words, the initial state is 0 at the initial time t, i.e.X(t) = 0, then we note X 0,u (•) is the adapted solution of the following system: (1.5) and the corresponding cost functional is as follows: For i = 1, 2, . . ., d, then we make some assumptions on the coefficients.

Assumption 1.4. The coefficients of state equation
On the other hand, we can rewrite (1.3): Similarly, we also have (1.9) In addition, we impose the following assumption:

Existence and uniqueness of optimal control
Lemma 2.1.Under Assumptions 1.4-1.6,for any (t, x) ∈ [0, T ] × R n , we claim that the cost functional J(t, x; u(•)) is continuous over A.
Proof.By means of Lemma 1.2 in [18], for any admissible control u(•) ∈ A, the state equation (1.1) has a unique solution X(•).And by applying the Itô's formula to |X(s)| 2 , Gronwall inequality and B-D-G inequality, we can easily obtain that there is a constant K satisfy (2.1) Suppose that X(•) is the state process corresponding to another admissible control ū(•) ∈ A, similarly, by applying the Itô's formula to |X(s) − X(s)| 2 , we can conclude: Combining with (1.3), and making use of Hölder inequality and the elementary inequality 2xy ≤ x 2 + y 2 , ∀x, y > 0, <∞. ( After applying Hölder inequality, we have the following estimate: R(s)u(s), u(s) − R(s)ū(s), ū(s) , S(s) u(s), X(s) − S(s) ū(s), X(s) can be treated in the same way, and the mean-field type is also applicable, then we have: (2.5) Thus, we can conclude that where X(•) or X x,u (•) is the solution of (1.1) corresponding to the admissible control u(•) and X 0,v (•) is the solution of (1.5) corresponding to the admissible control v(•).
Proof.Since the state equation is linear, for ∀ε ∈ (0, 1), it is easy to check that After making use of (2.7), and subtracting from (1.3), we obtain we express the right hand of (2.6) as ∆ u,v .Specially, when we set ε = 1, combining with (1.6), it's easy to verify that On the other hand, according to (2.3), the following estimate holds, Therefore, by the definition of Fréchet differentiable, lim which implies that the cost functional J(t, x; u(•)) has Fréchet derivative J (t, x; u(•)) given by (2.6).
Proof.In the process of proving Lemma 2.2, we have already got (2.8), comparing with (1.6) and (2.6), (2.13) On the other hand, according to (2.13), we can easily obtain that lim (2.14) Thus, we can conclude that since the cost functional J(t, x; u(•)) is Fréchet differentiable, then it is also Gâteaux differentiable, and the Gâteaux derivative is the same as the Fréchet derivative.
Proof.According to Lemma 2.1-2.4,we have already illustrated the continuity, coercive, strictly convexity and Fréchet differentiability of J(t, x; u(•)).Then we can obtain directly that the optimal control of Problem 1.1 exists and is unique according to the Proposition 2.1.2 of [4].
Theorem 2.6.Under Assumptions 1.4-1.6,for any admissible control v(•) ∈ A, the necessary and sufficient condition for an admissible control u(•) ∈ A to be an optimal control of Problem 1.1 is that Proof.(Necessary): If u * (•) is the optimal control, according to Problem 1.1, we have from the definition of Gâteaux derivative J (t, x; u(•)), we can get then we deduce that (Sufficient): If there exists a control u(•) ∈ A, s.t.J (t, x; u(•)), v(•) = 0 holds, we seek to prove that u(•) is the optimal control.In other words, we just need to prove for any admission control v(•) ∈ A, Actually, according to (2.12) and (2.18), we have which implies that u(•) is the optimal control.The proof is complete.
Proof.Under partial information, we just need to apply conditional mathematics expectation E[•|G s ] to (4.25).

Existence and uniqueness of Riccati equations
Under Assumptions 1.4-1.6,according to the Theorem 4.1, we know that the solvability of Problem 1.1 is equivalent to the solvability of the two integro-differential Riccati equations (4.19) and (4.20).This part we will discuss the existence and uniqueness the solutions of Riccati equations (4.19) and (4.20) under standard Assumptions 1.4-1.6.To prove this theorem, we need the following lemma.
Since the integro-differential Riccati equation (4.19) is completely nonlinear, we can first make it liner in formal by Quasi-linearization and then get the existence and uniqueness of the solution by the principle of compression mapping and quasi-linearization method.To this end, we need the following proposition:

Conclusion
This paper mainly studies a kind of MF-LQ optimal control problem with jumps under partial information.Through the adjoint processes, Hamiltonian system and filtering technique, we deduce two integro-differential Riccati equations, shown that the optimal control can be represented as a feedback form under complete and partial information.And by means of discuss the existence and uniqueness of Riccati equations, we conclude the feedback representation of optimal control of MF-LQ problem is unique.Finally, we explore a special case, get the corresponding Hamiltonian system, the representation of optimal control and Riccati equations.