Optimality Conditions for Optimal Control of Jump-Diffusion SDEs with Correlated Observations Noises

This paper is concerned with necessary and sufficient optimality conditions for optimal control of jump-diffusion stochastic differential equations. Compared with the existing literature, there are two distinguishing features: one is that the states are driven by Brownianmotions and Poisson randommeasure; the other one is that the states and the observations are correlated.We derive a necessary and a sufficient conditions in the form ofmaximumprinciple when control domain is convex. A linear-quadratic example is worked out to illustrate the applications of the foregoing optimality conditions.


Introduction
The purpose of this paper is to establish maximum principle, also called necessary optimality conditions, for optimal control of jump-diffusion stochastic differential equations driven by Brownian motions and Poisson random measure where states and observations are correlated.There is rich literature on maximum principle for optimal control of stochastic differential equations (SDEs, for short).For example, Peng [1] proved a general stochastic maximum principle for optimal control of diffusion SDEs by introducing second-order variational equations; Tang and Li [2] derived a general necessary condition for optimal control of jump-diffusion SDEs; Pham [3] gave a survey on some recent aspects and developments in stochastic control of diffusion processes and discussed the main historical approaches and their modern exposition for studying stochastic control problems.In many situations, the states of the systems cannot be completely observed; however, some other processes related to the unobservable states can be observed.Then partially observable optimal control of SDEs also attracts much research attention.Such a subject has been discussed by many authors, such as Baras et al. [4], Bensoussan [5], Fleming [6], Li and Tang [7], Tang and Hou [8], Huang et al. [9], Wang and Wu [10,11], Xiao [12], Zhou [13], and Hu and Øksendal [14].
In the real world, there usually exists certain correlated noises between the state and observation which is more general case than the case of independent noises.However, the literature mentioned above only deals with the case that states and observations are independent.To the author's best knowledge, there are only two papers about correlated observation noises.Tang [15] considered optimal control of diffusion SDEs and proved a general stochastic maximum principle; Xiao [16] obtained maximum principle for optimal control of Poisson point processes with correlated Gaussian white noisy observations.Up till now, there is no literature on optimal control of jump-diffusion SDEs with correlated observation noises.Here, we set out to study the optimal control problem of SDEs driven by Brownian motions and Poisson random measure in the case of correlated noisy observations.Due to the correlation between the state and the observation, there exists a weak solution rather than a strong solution to the state equation, which is different from the case of noncorrelated noisy observation.Meanwhile, the superfluous adjoint processes arise in the optimality condition, and we need some complicated matrices decomposition to eliminate them.We also make an effort to seek a suitable adjoint of the drift coefficient of the observation equation which plays an important role in defining the new Hamiltonian function, so that the improved necessary maximum principle is just the thing we want.
The rest of this paper is organized as follows.In Section 2, we formulate the optimal control problem of partially observable jump-diffusion SDEs with correlated observation noises.
In Section 3, we obtain the partially observed stochastic maximum principle when the control domain is convex.Section 4 presents a linear-quadratic example to illustrate the applications of the theoretical results derived in Section 3. Some conclusions are given in Section 5.

Formulation of Problem
Throughout this paper, we assume that E is a nonempty Borel subset of R  , B(E) is the Borel -algebra generated by E, and (⋅) is a -finite measure on (E, B(E)).Let  > 0 be a fixed real number.Let (⋅) be a stationary F  -Poisson point process on E with the characteristic measure ().We denote by ( ) the counting measure or Poisson measure induced by (⋅) and set Ñ( ) = ( ) − () satisfying In general, we call () and Ñ( ) the intensity and the compensated Poisson measure of ( ), respectively (see Ikeda and Watanabe [17]).In addition, let (Ω, F, ) be a complete probability space on which two mutually independent standard Brownian motions (⋅) and (⋅) are defined, valued in R  and R  , respectively, independent of (⋅).Let F   , F   , and F   be the natural filtration generated by (⋅), (⋅), and (⋅), respectively.We assume that where N denotes the totality of -null sets.For a matrix, we use superscripts to indicate (when necessary) the number of its columns or its rows or the position of its components, and precise meaning can be specified from the context; the range of the superscripts will not be explicitly stated unless there is a danger of confusion.
Let  be a nonempty convex subsets of some Euclidean space.A control is a stochastic process  : Ω × [0, ] → .For notational simplicity, hereinafter, we will omit  in random functions.We define an admissible control set U ad by Every element of U ad is called an admissible control.
Introduce the mappings  : We make the following hypothesis.(H1) , , and  are continuously differentiable with respect to (, V).They are bounded by (1 + || + |V|) and their derivatives with respect to (, V) are uniformly bounded. and ℎ are uniformly bounded and continuously differentiable with respect to (, V), whose derivatives with respect to (, V) are also uniformly bounded. and  are continuously differentiable with respect to (, V).There exists a constant  0 such that The partially observable optimal control problem is stated as follows.
Consider the state and the observation where  V (⋅) is a stochastic process depending on the control V(⋅).Note that if the diffusion term  ̸ = 0 in (3), then there exist the correlated noise  V between the state and observation.Substituting (4) into (3), we have For each V(⋅) ∈ U ad , there exists a unique strong solution to (5) (see Ikeda and Watanabe [17], which will be denoted by that is and  V :=  V ().Note that since ℎ is uniformly bounded, we actually have that  V () is a martingale on (Ω, F, ), then the expression  V :=  V () is to say that  V ≪ , that is,  V is absolutely continuously w.r.t..Note that  V () > 0 a.s., so we also have that  ≪  V .Hence the two measures  V and  are equivalent.Furthermore, Therefore  V is also a probability measure.From Girsanov's theorem, it follows that ( V ,  V , ,  V , Ñ, ) is a weak solution on (Ω, F, F  ) to ( 3) and ( 4).Throughout this paper, the superscript symbol * means the transpose of certain vector or matrix.Consider the cost functional Here, E V denotes the expectation with respect to the probability space (Ω, F,  V ).Our partially observed optimal control problem is to minimize the cost functional (8) over V(⋅) ∈ U ad , subject to (3) and ( 4).Obviously, the cost functional (8) can be rewritten as So the original optimization problem is equivalent to minimizing the cost functional (9) over V(⋅) ∈ U ad , subject to ( 5) and (7).
If an admissible control (⋅) minimizes the cost functional (if it does exist), then it is called optimal and the corresponding weak solution (  , , , , Ñ, ) to ( 3) and ( 4) is called the optimal trajectory.

Maximum Principle
In this section, we shall establish a necessary and a sufficient maximum principles.
We note that minimizing the cost functional (9) over V(⋅) ∈ U ad , subject to ( 5) and (7), is similar to a completely observable optimal control problem except for the difference of admissible control set.It is a heuristic method to derive the maximum principle for partially observable optimal control from the maximum principle for completely observable optimal control.We now specify this point.
Let (⋅) be an optimal control and (  , , , , Ñ, ) the corresponding optimal trajectory.Set Equations ( 5) and ( 7) can be compressed into the following form: The cost function ( 9) is rewritten as Our partially observed optimal control problem becomes the following minimization problem: to minimize (V(⋅)) in (12) over V(⋅) ∈ U ad , subject to (11).The present formulation of the partially observable optimal control problem is quite similar to a completely observed optimal control problem; the only difference lies in the admissible controls class U ad .We can follow the same arguments in the case of full information to derive the following desired maximum principle (see Tang and Li [2] in the case of complete information and convex control domain).
Define the Hamiltonian  :

Mathematical Problems in Engineering
For simplicity, we introduce the notation  (,  ()) ≐  (,  () ,  () ,  () ,  () ,  () ,  ()) .( From the maximum principle in Tang and Li [2], we obtain the following necessary maximum principle for partially observable optimal control.Lemma 1. Assume that the hypothesis (H1) holds.Let (⋅) be an optimal control and (  , , , , Ñ, ) the optimal trajectory to (3) and (4).Then We note that since the variable (⋅), which is regarded as the state variable for a moment, appears in the optimal control problem in a linear way, some adjoint processes are superfluous in the above maximum principle (16).Inspired by the method in Tang [15], we set about dispensing with these adjoint processes and reformulate the above maximum principle.
Subsequently, we set out to derive the sufficient optimality conditions for the foregoing optimal control problem.We introduce the following assumption.
(H2) (⋅) is convex in .Function ℎ is independent of variables  and V.
Inspired by Xiao [12] and Huang et al. [18], we have the following theorem.

An LQ Example
To illustrate that the foregoing theories may find the interesting applications in practice, we work out an LQ example of partially observable optimal control of jump-diffusion system with correlated noisy observations.Firstly, by applying the necessary maximum principle, we find a candidate optimal control.Then by sufficient maximum principle, we verify that it is indeed an optimal control.Finally, by certain techniques of forward-backward stochastic differential equations filtering, we obtain an explicit expression of the optimal control.Consider a partially observable 1-dimensional control system with the observation and the cost functional Here, the coefficients   ( = 1, . . ., 5) and  are bounded and deterministic.The set of admissible controls is defined by E V denotes the expectation with respect to the probability space (Ω, F,  V ),  V =  V () and We aim to find an explicitly optimal control to minimize the cost functional (V(⋅)) over V(⋅) ∈ U ad , subject to (33) and (34).Now we begin to seek the explicit optimal control by three steps.First Step.Find candidate optimal controls.
We firstly write down the Hamiltonian function where  is the trajectory to (33) corresponding to the candidate optimal control (⋅).By Theorem 2, we find a unique candidate optimal control (⋅) which satisfies the following expression: (40)

Second
Step.Verify (⋅) in ( 39) is indeed optimal.We can check that all conditions in Theorem 3 are satisfied, so (⋅) in (39) is indeed optimal.Third Step.Give explicit expression of optimal control.
Although we obtain its nominal form, the expression of optimal control (⋅) in (39) is not quite explicit and satisfactory.From (33), we know that the state (⋅) is dependent on the control (⋅).Since (⋅) and (⋅) are coupled at time  and the control (⋅) in (39) is dependent on the adjoint state (⋅), (33) and (40) compose mutually coupled forwardbackward stochastic systems which makes it difficult to find their explicit solution.Next, we try to get a more explicit and observable expression than the one in (39) by certain filtering techniques.

Conclusion Remark
This paper has investigated the optimal control problem of partially observable jump-diffusion SDEs.The most distinguishing feature is, compared with the existing literature, that the states and observations are correlated.By transforming the partial observation problem to a related problem with full information, we established a necessary and a sufficient optimality conditions.Under the framework of convex control domain, our maximum principle can cover Tang and Hou [8], Tang [15] and Xiao [12]) as particular cases, but does not establish the relations among the adjoint processes.In the future work, we shall introduce some adjoint vector fields to characterize the adjoint processes and derive some other formulations of partially observable stochastic maximum principle.