Mean-Field Stochastic Linear-Quadratic Optimal Controls: Roles of Expectation and Conditional Expectation Operators

TL;DR

This paper explores a mean-field linear-quadratic control problem involving expectation and conditional expectation, deriving solutions and analyzing how these operators affect time-consistency and Riccati equation well-posedness.

Contribution

It explicitly derives solutions for different control strategies and clarifies the roles of expectation operators in the problem's time-consistency.

Findings

Explicit solutions for pre-committed, naive, and equilibrium controls

Analysis of how expectation operators influence time-consistency

Establishment of Riccati equations' well-posedness

Abstract

This paper investigates a mean-field linear-quadratic optimal control problem where the state dynamics and cost functional incorporate both expectation and conditional expectation terms. We explicitly derive the pre-committed, na\"{\i}ve, and equilibrium solutions and establish the well-posedness of the associated Riccati equations. This reveals how the expectation and conditional expectation operators influence time-consistency.