# Linear Quadratic Stochastic Optimal Control Problems with Operator Coefficients: Open-Loop Solutions

## TL;DR

This paper studies linear quadratic stochastic optimal control problems with operator coefficients, focusing on open-loop solutions and their relation to coupled forward-backward stochastic differential equations, especially in mean-field contexts.

## Contribution

It establishes conditions for the well-posedness of operator-coefficient FBSDEs and applies these results to solve general mean-field linear quadratic control problems.

## Key findings

- Established well-posedness of operator-coefficient FBSDEs.
- Proved existence of open-loop optimal controls under certain conditions.
- Solved a general mean-field LQ control problem using the developed theory.

## Abstract

An optimal control problem is considered for linear stochastic differential equations with quadratic cost functional. The coefficients of the state equation and the weights in the cost functional are bounded operators on the spaces of square integrable random variables. The main motivation of our study is linear quadratic optimal control problems for mean-field stochastic differential equations. Open-loop solvability of the problem is investigated, which is characterized as the solvability of a system of linear coupled forward-backward stochastic differential equations (FBSDE, for short) with operator coefficients. Under proper conditions, the well-posedness of such an FBSDE is established, which leads to the existence of an open-loop optimal control. Finally, as an application of our main results, a general mean-field linear quadratic control problem in the open-loop case is solved.