An optimal control problem for linear SDE of mean-field type with terminal constraint and partial information

Haiyan Zhang

doi:10.1186/s13662-019-2029-0

Abstract

This paper is concerned with an optimal control problem for a linear stochastic differential equation (SDE) of mean-field type, where the drift coefficient of observation equation is linear with respect to the state, the control and their expectations, and the state is subject to a terminal constraint. The control problem cannot be solved by transforming it into a standard optimal control problem for an SDE without mean-field term. By virtue of a backward separation method with a decomposition technique, one optimality condition and one forward–backward filter are derived. Two linear-quadratic (LQ) optimal control problems and one cash management problem with terminal constraint and partial information are studied, and optimal feedback controls are explicitly obtained.

Highlights

One begins with a complete filtered probability space (Ω, F, (Ft)0≤t≤T, P), on which are given an Ft-adapted standard Brownian motion with value in R2 and a Gaussian random variable ξ with mean μ0 and covariance σ0. (ω, ω ) is independent of ξ
Combining the backward separation method with Girsanov’s measure transformation, the circular dependence between v and yv was decoupled in Wang et al [3], and a necessary condition for optimality was derived
The condition together with forward–backward filter provides an effective method for studying stochastic optimal control with terminal constraint and incomplete information

Summary

Introduction

One begins with a complete filtered probability space (Ω, F , (Ft)0≤t≤T , P), on which are given an Ft-adapted standard Brownian motion (ωt, ωt) with value in R2 and a Gaussian random variable ξ with mean μ0 and covariance σ0. (ω, ω ) is independent of ξ. In 2018, Wang coauthored their monograph [2], where the backward separation method was systematically introduced and was regarded as one of most important tools for studying partially observed optimal control problems. Combining the backward separation method with Girsanov’s measure transformation, the circular dependence between v and yv was decoupled in Wang et al [3], and a necessary condition for optimality was derived Along this line, Zhang [4], Ma and Liu [5] extended [3] to the case of correlated state and observation noises, and the case of risk-sensitive control, respectively. The condition together with forward–backward filter provides an effective method for studying stochastic optimal control with terminal constraint and incomplete information.

Problem formulation and preliminary Define x0 and y0 by two SDEs

Concluding remarks