A Class of Discrete-time Mean-field Stochastic Linear-quadratic Optimal Control Problems with Financial Application

TL;DR

This paper addresses a discrete-time mean-field stochastic linear-quadratic control problem from finance, deriving optimal controls via Riccati equations and operator methods, with practical financial application demonstrated.

Contribution

It introduces a novel operator-based reformulation and solution approach for mean-field stochastic LQ control problems, linking matrix optimization with Riccati equations in a financial context.

Findings

Optimal control characterized by six algebraic Riccati difference equations

Operator reformulation aligns with classical completing the square method

Application to a multidimensional noise financial example

Abstract

This paper is concerned with a discrete-time mean-field stochastic linear-quadratic optimal control problem arose from financial application. Through matrix dynamical optimization method, a group of linear feedback controls is investigated. The problem is then reformulated as an operator stochastic linear-quadratic optimal control problem by a sequence of bounded linear operators over Hilbert space, the optimal control with six algebraic Riccati difference equations is obtained by backward induction. The two above approaches are proved to be coincided by the classical method of completing the square. Finally, after discussing the solution of the problem under multidimensional noises, a financial application example is given.