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We study the partial information classical and impulse controls problem of forward-backward systems driven by Lévy processes, where the control variable consists of two components: the classical stochastic control and the impulse control; the information available to the controller is possibly less than the full information, that is, partial information. We derive a maximum principle to give the sufficient and necessary optimality conditions for the local critical points of the classical and impulse controls problem. As an application, we apply the maximum principle to a portfolio optimization problem with piecewise consumption processes and give its explicit solutions.

The classical and impulse controls problems have received considerable attention in recent years due to their wide applicability in different areas, such as optimal control of the exchange rate between different currencies (see, e.g., [

In the existing literatures, the dynamic programming principle and the maximum principle are two main approaches in solving these problems.

In dynamic programming principle, the classical and impulse controls can be solved by a verification theorem and the value function is a solution to some quasi-variational inequalities. However, the dynamic programming approach relies on the assumption that the controlled system is Markovian; see, for example, [

There have been some pioneering works on deriving maximum principles for the classical and impulse controls problems. For example, Wu and Zhang [

In many practical systems, the controller only gets partial information, instead of full information, such as delayed information (see, e.g., [

In this paper, we study classical and impulse controls problems of forward-backward systems, where the stochastic systems are represented by forward-backward SDEs driven by Lévy processes, the control variable consists of two components: the stochastic control

The similar maximum principle is also studied by Wu and Zhang [

The paper is organized as follows: in the next section we formulate the partial information classical and impulse controls of the forward-backward system driven by Lévy processes. In Section

Let

Suppose that we are given a subfiltration

Let

Now we consider the forward-backward systems involving classical and impulse controls. Given

There are two different jumps in the system (

A jump

Let

Suppose we are given a performance functional of the form

In this section, we derive a maximum principle for the optimal control problems (

Firstly, we make the following assumptions.

(1) For all

(2) For all

Next we give the definition of the Hamiltonian process.

We define a Hamiltonian process

(i) Forward system in the unknown process

(ii) Backward system in the unknown processes

For the sake of simplicity, we use the short hand notation in the following:

Let

(1)

(2) Consider

Define

Firstly, we prove

Now we prove that (

Let

Let

In a financial market, we are given a subfiltration

Endowed with initial wealth

The control problem (

If

On the other hand, by the sufficient and necessary optimality condition (

Let

We consider the partial information classical and impulse controls problem of forward-backward systems driven by Lévy processes. The control variable consists of two components: the classical stochastic control and the impulse control. Because of the non-Markovian nature of the partial information, dynamic programming principle cannot be used to solve partial information control problems. As a result, we derive a maximum principle for this partial information problem. Because the concavity conditions of the utility functions and the Hamiltonian process may not hold in many applications, we give the sufficient and necessary optimality conditions for the local critical points of the control problem. To illustrate the theoretical results, we use the maximum principle to solve a portfolio optimization problem with piecewise consumption processes and give its explicit solutions.

In this paper, we assume that the two different jumps in our system do not occur at the same time (Assumption

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China under Grant 11171050.