Linear Quadratic Control of Backward Stochastic Differential Equation with Partial Information

TL;DR

This paper develops explicit solutions for optimal control of linear backward stochastic differential equations with quadratic costs under partial information, using stochastic maximum principle and decoupling techniques.

Contribution

It provides a complete explicit solution framework for the control problem, including Riccati equations and feedback control, under partial information.

Findings

Explicit optimal control formulas derived

Three Riccati equations identified and solved

Numerical simulations demonstrate effectiveness

Abstract

In this paper, we study an optimal control problem of linear backward stochastic differential equation (BSDE) with quadratic cost functional under partial information. This problem is solved completely and explicitly by using a stochastic maximum principle and a decoupling technique. By using the maximum principle, a stochastic Hamiltonian system, which is a forward-backward stochastic differential equation (FBSDE) with filtering, is obtained. By decoupling the stochastic Hamiltonian system, three Riccati equations, a BSDE with filtering, and a stochastic differential equation (SDE) with filtering are derived. We then get an optimal control with a feedback representation. An explicit formula for the corresponding optimal cost is also established. As illustrative examples, we consider two special scalar-valued control problems and give some numerical simulations.