Control Problem Of Mean-field Type Research Articles

Maximum Principle for Mean Field Type Control Problems with General Volatility Functions

In this paper, we study the maximum principle of mean field type control problems when the volatility function depends on the state and its measure and also the control, by using our recently developed method in [Bensoussan, A., Huang, Z. and Yam, S. C. P. [2023] Control theory on Wasserstein space: A new approach to optimality conditions, Ann. Math. Sci. Appl.; Bensoussan, A., Tai, H. M. and Yam, S. C. P. [2023] Mean field type control problems, some Hilbert-space-valued FBSDEs, and related equations, preprint (2023), arXiv:2305.04019; Bensoussan, A. and Yam, S. C. P. [2019] Control problem on space of random variables and master equation, ESAIM Control Optim. Calc. Var. 25, 10]. Our method is to embed the mean field type control problem into a Hilbert space to bypass the evolution in the Wasserstein space. We here give a necessary condition and a sufficient condition for these control problems in Hilbert spaces, and we also derive a system of forward–backward stochastic differential equations.

Classical solutions to local first-order extended mean field games

We study the existence of classical solutions to a broad class of local, first order, forward-backward extended mean field games systems, that includes standard mean field games, mean field games with congestion, and mean field type control problems. We work with a strictly monotone cost that may be fully coupled with the Hamiltonian, which is assumed to have superlinear growth. Following previous work on the standard first order mean field games system, we prove the existence of smooth solutions under a coercivity condition that ensures a positive density of players, assuming a strict form of the uniqueness condition for the system. Our work relies on transforming the problem into a partial differential equation with oblique boundary conditions, which is elliptic precisely under the uniqueness condition.

Solvability of Infinite Horizon McKean–Vlasov FBSDEs in Mean Field Control Problems and Games

In this paper, we show existence and uniqueness of solutions of the infinite horizon McKean–Vlasov FBSDEs using two different methods, which lead to two different sets of assumptions. We use these results to solve the infinite horizon mean field type control problems and mean field games.

Maximum principle for discrete-time stochastic control problem of mean-field type

In this work, a discrete-time mean-field type stochastic optimal control problem is studied. The goal is to derive the stochastic maximum principle with convex control domains. L-derivative is applied to handle the mean-field term and a technique of adjoint operator is used to overcome the difficulties of obtaining adjoint equations and duality relation. Then, the stochastic maximum principle for discrete-time mean-field type stochastic optimal control problem is established. Finally, as an illustration of our stochastic maximum principle, a discrete-time mean–variance portfolio selection problem is solved with the decoupling technique which is different from the continuous-time case.

A mean-field stochastic linear-quadratic optimal control problem with jumps under partial information

In this article, the stochastic linear-quadratic optimal control problem of mean-field type with jumps under partial information is discussed. The state equation which contains affine terms is a SDE with jumps driven by a multidimensional Brownian motion and a Poisson stochastic martingale measure, and the quadratic cost function contains cross terms. In addition, the state and the control as well as their expectations are contained both in the state equation and the cost functional. This is the so-called optimal control problem of mean-field type. Firstly, the existence and uniqueness of the optimal control is proved. Secondly, the adjoint processes of the state equation is introduced, and by using the duality technique, the optimal control is characterized by the stochastic Hamiltonian system. Thirdly, by applying a decoupling technology, we deduce two integro-differential Riccati equations and get the feedback representation of the optimal control under partial information. Fourthly, the existence and uniqueness of the solutions of two Riccati equations are proved. Finally, we discuss a special case, and establish the corresponding feedback representation of the optimal control by means of filtering technique.

Об управлении вероятностными потоками в условиях неопределенности

The mean-field type control problems with incomplete information are considered. There are several points of view that can be adopted to study the dynamics in probability space. Eulerian framework describes probability flows by the continuity equation. Kantorovich formulation describes each probability flows in terms of a single distribution on the set of admissible trajectories. The superposition principle connects these frameworks for uncontrolled dynamics. In this article, a probability flow in the both frameworks must be generated by a control that based on incomplete information about state and/or the probability at every time instance. This article presents some links between these frameworks in the case of incomplete information. In particular, besides the convexity condition, the assumptions are founded that guarantees the equivalence between the Kantorovich and Eulerian framework. This expands [6, Theorem 1] to mean-field type control problem with incomplete information.

Mean-field stochastic H 2/H ∞ control with delay

The finite horizon control problem of mean-field type for delayed stochastic systems is considered in this paper. Firstly, we derive a sufficient mean-field generalised stochastic bounded real lemma (SBRL). Secondly, based on the generalized SBRL and the mean-field anticipated forward-backward stochastic delay differential equations, this paper establishes a sufficient condition for the existence of a solution to delayed stochastic control of mean-field type. Thirdly, optimal state feedback regulators are studied for the case with delays only in control variable and disturbance signal via the solvability of coupled matrix-valued equations. Finally, an example is employed to show the effectiveness of the results obtained.

Mean-field type quantum filter for a quantum Ising type system

Mean-field games or mean-field type control problems are one of distributed control schemes that reduce the computational burden. In this paper, a quantum version of mean-field game settings is developed and the mean-field type quantum filter is derived for quantum Ising models.

Forward and Backward Mean-Field Stochastic Partial Differential Equation and Optimal Control

This paper is mainly concerned with the solutions to both forward and backward mean-field stochastic partial differential equation and the corresponding optimal control problem for mean-field stochastic partial differential equation. The authors first prove the continuous dependence theorems of forward and backward mean-field stochastic partial differential equations and show the existence and uniqueness of solutions to them. Then they establish necessary and sufficient optimality conditions of the control problem in the form of Pontryagin’s maximum principles. To illustrate the theoretical results, the authors apply stochastic maximum principles to study the infinite-dimensional linear-quadratic control problem of mean-field type. Further, an application to a Cauchy problem for a controlled stochastic linear PDE of mean-field type is studied.

Control problem on space of random variables and master equation

In this article, we study a control problem in an appropriate space of random variables; in fact, in our set up, we can consider an arbitrary Hilbert space, yet we specialize only to a Hilbert space of square-integrable random variables. We see that the control problem can then be related to a mean field type control problem. We explore here a suggestion of Lions in (Lectures at College de France, http://www.college-de-france.fr) and (Seminar at College de France). Mean field type control problems are control problems in which functionals depend on probability measures of the underlying controlled process. Gangbo and Święch [J. Differ. Equ. 259 (2015) 6573–6643] considered this type of problem in the space of probability measures equipped with the Wasserstein metric and use the concept of Wasserstein gradient; their work provides a completely rigorous treatment, but it is quite intricate, because metric spaces are not vector spaces. The approach suggested by Lions overcomes this difficulty. Nevertheless, our present proposed approach also benefits from the useful concept of L-derivatives as introduced in a recent interesting treatise of Carmona and Delarue [Probabilistic Theory of Mean Field Games with Applications. Springer Verlag (2017)]. We also consider Bellman equation and the Master equation of mean field type control. We provide also some extension of the results of Gangbo and Święch [J. Differ. Equ. 259 (2015) 6573–6643].

Probabilistic Approach to Mean Field Games and Mean Field Type Control Problems with Multiple Populations

In this work, we systematically investigate mean field games and mean field type control problems with multiple populations using a coupled system of forward-backward stochastic differential equations of McKean-Vlasov type stemming from Pontryagin's stochastic maximum principle. Although the same cost functions as well as the coefficient functions of the state dynamics are shared among the agents within each population, they can be different population by population. We study the mean field limit for the three different situations; (i) every agent is non-cooperative; (ii) the agents within each population are cooperative; and (iii) the agents in some populations are cooperative but those in the other populations are not. We provide several sets of sufficient conditions for the existence of a mean field equilibrium for each of these cases. Furthermore, under appropriate conditions, we show that the mean field solution to each of these problems actually provides an approximate Nash equilibrium for the corresponding game with a large but finite number of agents.

Discrete-time mean-field stochastic H 2/H ∞ control

The finite horizon H 2/H ∞ control problem of mean-field type for discrete-time systems is considered in this paper. Firstly, the authors derive a mean-field stochastic bounded real lemma (SBRL). Secondly, a sufficient condition for the solvability of discrete-time mean-field stochastic linearquadratic (LQ) optimal control is presented. Thirdly, based on SBRL and LQ results, this paper establishes a sufficient condition for the existence of discrete-time stochastic H 2/H ∞ control of meanfield type via the solvability of coupled matrix-valued equations.

On partial-information optimal singular control problem for mean-field stochastic differential equations driven by Teugels martingales measures

ABSTRACTThis paper is concerned with partial-information mixed optimal stochastic continuous–singular control problem for mean-field stochastic differential equation driven by Teugels martingales and an independent Brownian motion, where the Teugels martingales are a family of pairwise strongly orthonormal martingales associated with Lévy processes. The control variable has two components; the first being absolutely continuous, and the second singular. Partial-information necessary and sufficient conditions of optimal continuous–singular control for these mean-field models are investigated. As an illustration, this paper studies a partial-information linear quadratic control problem of mean-field type involving continuous–singular control.

Dynamic Programming for Mean-Field Type Control

We investigate a model problem for optimal resource management. The problem is a stochastic control problem of mean-field type. We compare a Hamilton---Jacobi---Bellman fixed-point algorithm to a steepest descent method issued from calculus of variations. For mean-field type control problems, stochastic dynamic programming requires adaptation. The problem is reformulated as a distributed control problem by using the Fokker---Planck equation for the probability distribution of the stochastic process; then, an extended Bellman's principle is derived by a different argument than the one used by P. L. Lions. Both algorithms are compared numerically.

Dynamic programming for mean-field type control

For mean-field type control problems, stochastic dynamic programming requires adaptation. We propose to reformulate the problem as a distributed control problem by assuming that the PDF ρ of the stochastic process exists. Then we show that Bellman's principle applies to the dynamic programming value function V(τ,ρτ), where the dependency on ρτ is functional as in P.-L. Lions' analysis of mean-field games (2007) [10]. We derive HJB equations and apply them to two examples, a portfolio optimization and a systemic risk model.

A Mean-Field Necessary and Sufficient Conditions for Optimal Singular Stochastic Control

This paper studies singular optimal control problems for systems described by nonlinear-controlled stochastic differential equations of mean-field type (MFSDEs in short), in which the coefficients depend on the state of the solution process as well as of its expected value. Moreover, the cost functional is also of mean-field type. The control variable has two components, the first being absolutely continuous and the second singular. We establish necessary as well as sufficient conditions for optimal singular stochastic control where the system evolves according to MFSDEs. These conditions of optimality differs from the classical one in the sense that here the adjoint equation turns out to be a linear mean-field backward stochastic differential equation. The proof of our result is based on convex perturbation method of a given optimal control. The control domain is assumed to be convex. A linear quadratic stochastic optimal control problem of mean-field type is discussed as an illustrated example.

Stochastic maximum principle in the mean-field controls

In Buckdahn, Djehiche, Li, and Peng (2009), the authors obtained mean-field Backward Stochastic Differential Equations (BSDEs) in a natural way as a limit of some highly dimensional system of forward and backward SDEs, corresponding to a great number of “particles” (or “agents”). The objective of the present paper is to deepen the investigation of such mean-field BSDEs by studying their stochastic maximum principle. This paper studies the stochastic maximum principle (SMP) for mean-field controls, which is different from the classical ones. This paper deduces an SMP in integral form, and it also gets, under additional assumptions, necessary conditions as well as sufficient conditions for the optimality of a control. As an application, this paper studies a linear quadratic stochastic control problem of mean-field type.