Mean-field stochastic linear quadratic optimal control problems: closed-loop solvability

Xun Li,Jingrui Sun,Jiongmin Yong

doi:10.1186/s41546-016-0002-3

Abstract

An optimal control problem is studied for a linear mean-field stochastic differential equation with a quadratic cost functional. The coefficients and the weighting matrices in the cost functional are all assumed to be deterministic. Closed-loop strategies are introduced, which require to be independent of initial states; and such a nature makes it very useful and convenient in applications. In this paper, the existence of an optimal closed-loop strategy for the system (also called the closed-loop solvability of the problem) is characterized by the existence of a regular solution to the coupled two (generalized) Riccati equations, together with some constraints on the adapted solution to a linear backward stochastic differential equation and a linear terminal value problem of an ordinary differential equation.

Highlights

Let (, F, F, P) be a complete filtered probability space on which a standard onedimensional Brownian motion W = {W (t); 0 t < ∞} is defined, where F = {Ft }t 0 is the natural filtration of W augmented by all the P-null sets in F
We introduce the following cost functional: J (t, ξ ; u(·)) E GX(T ), X(T ) + 2 g, X(T ) + G E[X(T )], E[X(T )] + 2 g, E[X(T )]
In Section “Characterization of Closed-Loop Solvability”, we present our main result, in which the closed-loop solvability of the mean-field LQ problems is characterized

Summary

Introduction

Definition 2.1 Problem (MF-LQ) is said to be (uniquely) open-loop solvable at initial pair (t, ξ ) ∈ [0, T ] × L2Ft ( ; Rn) if there exists a (unique) u∗(·) ∈ U [t, T ] satisfying (1.3). (ii) For any ( (·), ̄ (·), v(·)) ∈ C [t, T ] and ξ ∈ L2Ft ( ; Rn), let X(·) ≡ X(· ; t, ξ, (·), ̄ (·), v(·)) be the solution to the following closed-loop system: If an optimal closedloop strategy (uniquely) exists on [t, T ], Problem (MF-LQ) is said to be (uniquely) closed-loop solvable on [t, T ].

Results

Conclusion