An optimal control problem for mean-field forward-backward stochastic differential equation with noisy observation

TL;DR

This paper studies an optimal control problem for mean-field forward-backward stochastic differential equations with noisy observations, providing new theoretical conditions and explicit solutions relevant to finance and risk management.

Contribution

It introduces a novel control framework with linear coefficients and derives optimality conditions and filters, expanding the application scope in financial mathematics.

Findings

Derived two coupled forward-backward optimal filters.

Established explicit solutions for linear-quadratic cases.

Extended the control theory for mean-field stochastic systems.

Abstract

This article is concerned with an optimal control problem derived by mean-field forward-backward stochastic differential equation with noisy observation, where the drift coefficients of the state equation and the observation equation are linear with respect to the state and its expectation. The control problem is different from the existing literature on optimal control for mean-field stochastic systems, and has more applications in mathematical finance, e.g., asset-liability management problem with recursive utility, systematic risk model. Using a backward separation method with a decomposition technique, two optimality conditions along with two coupled forward-backward optimal filters are derived. Several linear-quadratic optimal control problems for mean-field forward-backward stochastic differential equations are studied. Closed-form optimal solutions are explicitly obtained in detailed situations.