Social optima in mean field linear–quadratic–Gaussian models with control input constraint

Abstract

This paper studies linear quadratic social optimal problems with control input constraints, where both dynamics and costs involve the coupled term of states, and the diffusion terms contain control variables. In virtue of the variational method, we obtain the corresponding auxiliary stochastic control problem by introducing the limiting state-average process. Furthermore, we design a set of decentralized control strategies by using the stochastic maximum principle, which is given by a fully coupled non-linear mean-field forward–backward stochastic differential equation, and they are further shown to be asymptotically social optimality by perturbation analysis. As applications, we apply the theoretical results to deal with the optimal consumption problem and obtain the related optimal consumption strategy.

Introduction

In recent years, the study of mean-field game (MFG) and control has attracted consistent and extensive research attention because of their significant theoretical values and broad practical applications in various fields, such as economics, engineering, and management science (see, e.g., [1], [2], [3]). The MFG, a kind of non-cooperative game, involves a large number of agents (or players) in such stochastic systems, and each agent is insignificant and negligible, but their collective behaviors are significant. The methodology of MFG, as introduced by Lasry & Lions [4] and, independently, by Huang, Malhamé & Caines [5], has provided an effective and tractable analysis framework to establish the approximate Nash equilibrium for weakly-coupled stochastic controlled system with mean-field interactions. Interested readers may refer [6], [7], [8], [9] and the reference therein for more details of MFGs.

Besides non-cooperative games, social optimal problem, a type of cooperative game, in mean-field models has also drawn much attention (see [7], [10]). Here, all agents are cooperative and seek the same socially optimal decisions to minimize their common social cost, and its essence is a type of cooperate game. Therefore, in this case, to achieve the social optimal strategies, every agent should maintain a delicate balance between the reducing his own cost and the effect of the sum of all other agents’ costs. If each agent can use all agents’ information to find the socially optimal solution, the social optimal problem can be regarded as a standard high dimensional control problem, which can be dealt by the traditional control methods. However, when agents are quite numerous, this will bring considerably high computational complexity in a large-scale manner, and it is almost impossible to search and get all agents’ information. Therefore, one reasonable and practical direction is to investigate the related decentralized strategy based on local information only, which means that each agent can only make decisions by their own information. These methods are introduced by Huang, Caines & Malhamé [7] and used to provide an asymptotic team-optimal solution. After this, Huang & Nguyen consider a linear quadratic (LQ) mean-field control problem involving a major agent and a large number of minor agents [11], and Wang & Zhang investigate a mean-field social optimal problem with the Markov jump parameters [12]. For further literature, see [13] for the dynamic collective choice by finding a social optimum, and [14] for the social optimal problem in mean-field LQ control with volatility uncertainty. In addition, for more details, please refer [15], [16], [17].

In this paper, we study a class of social optimal problem in mean-field LQ control where the individual control domain is a convex subset of ${\mathbb{R}}^m$. We point out that, the control is unconstrained in all of the aforementioned works, and when we impose control constraints, the previous LQ approach fails to apply (see [18], [19]). There are a large number of applications, especially in finance and economics, in LQ control problems with control constraints. For example, the LQ control problems with positive control (i.e., the admissible control belong to the positive orthant) can be used to solve the no-shorting constraint in Markovitz optimal portfolio (see, e.g., [20], [21]); The optimal investment problems, where the investment should satisfy the linear portfolio constraints, which means that all investors can only invest in a part of stocks in the market, can also be addressed by dealing the corresponding LQ control problems with input constraints, and the interest readers can refer [22], [23]. In addition, there are a lot of famous works about MFGs with control input constraint, such as [24], [25]. As a response, in this paper, we consider the social optimal problem in mean-field form large-population systems with input constraints and achieve its financial applications.

The main contributions of this paper are as follows: Firstly, we propose and analyze a new type of mean-field LQ social optimal control problem with control input constraints, where both dynamics and costs involve the coupled term of states, and the diffusion terms can be dependent on their control variables, which bring many difficulties to construct the related decentralized control strategy. Secondly, the related Social Certainty Equivalence (SCE) system becomes a new type of nonlinear fully coupled mean-field forward–backward stochastic differential equation (MF-FBSDE) with projection operators. Since it contains nonlinear terms and its drift term involves adjoint processes, the previous approach used to deal with the SCE system fails to apply. Here, we establish its well-posedness in the global cases through the method of continuation under monotonic assumptions. Moreover, since the related SCE system is a fully coupled non-linear MF-FBSDE, to achieve the asymptotic social optimality, we should list more precise estimates about the dynamics and cost functionals based on the estimates of fully coupled nonlinear MF-FBSDEs, which have not been used before. Finally, a financial example, the optimal consumption problem, is introduced and solved, and the related financial explanations are also given.

The structure of this paper is as follows. Section 2 formulates the LQ social optimal problem with control input constraints. Section 3 is devoted to the social variational interpretation and gives the corresponding decentralized strategies. The proof of the asymptotic social optimality is given by Section 4. In Section 5, we give some examples and simulations to demonstrate the effectiveness of our obtained results. Finally, some conclusions and future research directions are given.

Notation. We denote the set of symmetric $n\times n$ matrices with real elements by ${\mathbb{S}}^n$. ${\mathbb{R}}^m$ denotes the $m$-dimensional Euclidean space with standard Euclidean norm $\left|\cdot \right|$ and standard Euclidean inner product $〈\cdot ,\cdot 〉$. The transpose of a vector (or matrix) $z$ is denoted by $z^{\top }$. The set of ${\mathbb{R}}^n$-valued continuous functions is denoted by $C([0,T];{\mathbb{R}}^n)$. If $N\left(\cdot \right)\in C([0,T];{\mathbb{S}}^n)$ and $N\left(t\right)\gg$ ($\ge$) $0$ for every $t\in [0,T]$, we say that $N\left(\cdot \right)$ is positive (semi) definite. Consider $h:[0,T]\times \Omega \to {\mathbb{R}}^n$ is an $\mathbb{F}\triangleq {\left\{{\mathcal{F}}_t\right\}}_{t\ge 0}$ adapted process. If $h$ is square integrable (i.e. $\mathbb{E}{\int }_0^T{\left|h\left(t\right)\right|}^{2}dt<\infty$), we shall write $h\in L_{\mathbb{F}}^2(0,T;{\mathbb{R}}^n)$; if $h$ is uniformly bounded (i.e., ${\text{esssup}}_{(t,\omega )\in [0,T]\times \Omega }\left|h\left(t\right)\right|<\infty$), then $h\in L_{\mathbb{F}}^{\infty }(0,T;{\mathbb{R}}^n)$. Furthermore, in cases where we are restricting ourselves to deterministic Borel measurable functions $h:[0,T]\to {\mathbb{R}}^n$, we shall drop the subscript $\mathbb{F}$ in the notation. Finally, we denote ${\Vert z\Vert }_Q^2\triangleq 〈Qz,z〉$, for all $Q\in {\mathbb{S}}^n$ and $z\in {\mathbb{R}}^n$.