Open-loop and closed-loop solvabilities for discrete-time mean-field stochastic linear quadratic optimal control problems

TL;DR

This paper investigates conditions for open-loop and closed-loop solvability in discrete-time mean-field stochastic LQ control problems with indefinite weights, providing characterizations via Riccati equations and convexity conditions.

Contribution

It introduces new criteria for solvability using generalized Riccati equations and convexity, extending existing theory to indefinite weighting matrices.

Findings

Open-loop solvability characterized by mean-field forward-backward stochastic difference equations.

Closed-loop solvability linked to solutions of generalized Riccati equations.

Finiteness of the control problem established through convexity analysis.

Abstract

This paper discusses the discrete-time mean-field stochastic linear quadratic optimal control problems, whose weighting matrices in the cost functional are not assumed to be definite. The open-loop solvability is characterized by the existence of the solution to a mean-field forward-backward stochastic difference equations with a convexity condition and a stationary condition. The closed-loop solvability is presented by virtue of the existences of the regular solution to the generalized Riccati equations and the solution to the linear recursive equation, which is also shown by the uniform convexity of the cost functional. Moreover, based on a family of uniformly convex cost functionals, the finiteness of the problem is characterized. Also, it turns out that a minimizing sequence, whose convergence is equivalent to the open-loop solvability of the problem. Finally, some examples are given to illustrate the theory developed.