Stochastic Maximum Principle for Partial Information Optimal Control Problem of Forward-Backward Systems Involving Classical and Impulse Controls

Yan Wang,Enmin Feng,Aimin Song

doi:10.1155/2014/452124

Yan Wang, Enmin Feng + Show 1 more

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https://doi.org/10.1155/2014/452124

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Abstract

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.

Highlights

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, optimal financing and dividend control problem of an insurance company facing fixed and proportional transaction costs, stochastic differential game, and dynamic output feedback controller design problem.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
We study the more general cases: the forward-backward system is driven by Levy processes and the information available to the controller is partial information
The control problem (38) is a classical and impulse controls problem of forward-backward systems driven by Levy processes under partial information Gt

Summary

Introduction

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., [1,2,3]), optimal financing and dividend control problem of an insurance company facing fixed and proportional transaction costs (see, e.g., [4, 5]), stochastic differential game (see, e.g., [6]), and dynamic output feedback controller design problem (see, e.g., [7] and the references therein). Wu and Zhang [11] established maximum principle for stochastic recursive optimal control problems involving impulse controls; Wu and Zhang [12] gave maximum principle for classical and impulse controls of forward-backward systems. In their control problems, the information available to the controller is full information. We study the more general cases: the forward-backward system is driven by Levy processes and the information available to the controller is partial information The paper is organized as follows: we formulate the partial information classical and impulse controls of the forward-backward system driven by Levy processes.

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