scholarly journals A Trading Execution Model Based on Mean Field Games and Optimal Control

2014 ◽  
Vol 05 (19) ◽  
pp. 3091-3116 ◽  
Author(s):  
Lorella Fatone ◽  
Francesca Mariani ◽  
Maria Cristina Recchioni ◽  
Francesco Zirilli
Keyword(s):  
Optimal Control ◽  
Mean Field ◽  
Mean Field Games ◽  
Model Based ◽  
Execution Model

scholarly journals Deep Learning and Mean-Field Games: A Stochastic Optimal Control Perspective

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 14
Author(s):  
Luca Di Persio ◽  
Matteo Garbelli
Keyword(s):  
Optimal Control ◽  
Deep Learning ◽  
Stochastic Optimal Control ◽  
Mathematical Formulation ◽  
Mean Field ◽  
Mean Field Games ◽  
Theoretical Frameworks ◽  
Depth Analysis ◽  
The Mean ◽  
Hamilton Jacobi Bellman

We provide a rigorous mathematical formulation of Deep Learning (DL) methodologies through an in-depth analysis of the learning procedures characterizing Neural Network (NN) models within the theoretical frameworks of Stochastic Optimal Control (SOC) and Mean-Field Games (MFGs). In particular, we show how the supervised learning approach can be translated in terms of a (stochastic) mean-field optimal control problem by applying the Hamilton–Jacobi–Bellman (HJB) approach and the mean-field Pontryagin maximum principle. Our contribution sheds new light on a possible theoretical connection between mean-field problems and DL, melting heterogeneous approaches and reporting the state-of-the-art within such fields to show how the latter different perspectives can be indeed fruitfully unified.

scholarly journals The Master Equation and the Convergence Problem in Mean Field Games

2019 ◽  
Author(s):  
Pierre Cardaliaguet ◽  
François Delarue ◽  
Jean-Michel Lasry ◽  
Pierre-Louis Lions
Keyword(s):  
Optimal Control ◽  
Master Equation ◽  
Differential Games ◽  
Mean Field ◽  
Mathematical Finance ◽  
Mean Field Games ◽  
Mean Field Limit ◽  
Continuum Of Players ◽  
Developing Area ◽  
Suitable Notion

This book describes the latest advances in the theory of mean field games, which are optimal control problems with a continuum of players, each of them interacting with the whole statistical distribution of a population. While it originated in economics, this theory now has applications in areas as diverse as mathematical finance, crowd phenomena, epidemiology, and cybersecurity. Because mean field games concern the interactions of infinitely many players in an optimal control framework, one expects them to appear as the limit for Nash equilibria of differential games with finitely many players as the number of players tends to infinity. The book rigorously establishes this convergence, which has been an open problem until now. The limit of the system associated with differential games with finitely many players is described by the so-called master equation, a nonlocal transport equation in the space of measures. After defining a suitable notion of differentiability in the space of measures, the authors provide a complete self-contained analysis of the master equation. Their analysis includes the case of common noise problems in which all the players are affected by a common Brownian motion. They then go on to explain how to use the master equation to prove the mean field limit. The book presents two important new results in mean field games that contribute to a unified theoretical framework for this exciting and fast-developing area of mathematics.

scholarly journals Minimal-time mean field games

2019 ◽  
Vol 29 (08) ◽  
pp. 1413-1464 ◽  
Author(s):  
Guilherme Mazanti ◽  
Filippo Santambrogio
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Value Function ◽  
Mean Field ◽  
Mean Field Games ◽  
Mean Field Game ◽  
Minimal Time ◽  
Regularity Properties ◽  
The Value Function

This paper considers a mean field game model inspired by crowd motion where agents want to leave a given bounded domain through a part of its boundary in minimal time. Each agent is free to move in any direction, but their maximal speed is bounded in terms of the average density of agents around their position in order to take into account congestion phenomena. After a preliminary study of the corresponding minimal-time optimal control problem, we formulate the mean field game in a Lagrangian setting and prove existence of Lagrangian equilibria using a fixed point strategy. We provide a further study of equilibria under the assumption that agents may leave the domain through the whole boundary, in which case equilibria are described through a system of a continuity equation on the distribution of agents coupled with a Hamilton–Jacobi equation on the value function of the optimal control problem solved by each agent. This is possible thanks to the semiconcavity of the value function, which follows from some further regularity properties of optimal trajectories obtained through Pontryagin Maximum Principle. Simulations illustrate the behavior of equilibria in some particular situations.

Stability Analysis in Mean-Field Games via an Evans Function Approach

2018 ◽  
Author(s):  
Piyush Grover
Keyword(s):  
Optimal Control ◽  
Stability Analysis ◽  
Mean Field ◽  
Evans Function ◽  
Function Approach ◽  
Mean Field Games ◽  
Mean Field Game ◽  
Large Populations ◽  
Multi Agent ◽  
Hamilton Jacobi Bellman

This work is concerned with stability analysis of stationary and time-varying equilibria in a class of mean-field games that relate to multi-agent control problems of flocking and swarming. The mean-field game framework is a non-cooperative model of distributed optimal control in large populations, and characterizes the optimal control for a representative agent in Nash-equilibrium with the population. A mean-field game model is described by a coupled PDE system of forward-in-time Fokker-Planck (FP) equation for density of agents, and a backward-in-time Hamilton-Jacobi-Bellman (HJB) equation for control. The linear stability analysis of fixed points of these equations typically proceeds via numerical computation of spectrum of the linearized MFG operator. We explore the Evans function approach that provides a geometric alternative to solving the characteristic equation.

scholarly journals A policy iteration method for Mean Field Games

2021 ◽  
Author(s):  
Simone Cacace ◽  
Fabio Camilli ◽  
Alessandro Goffi
Keyword(s):  
Optimal Control ◽  
Iteration Method ◽  
Optimal Control Problems ◽  
Mean Field ◽  
Policy Iteration ◽  
Mean Field Games ◽  
Control Problems ◽  
Discrete Problem ◽  
Classical Algorithm ◽  
Dimension One

The policy iteration method is a classical algorithm for solving optimal control problems. In this paper, we introduce a policy iteration method for Mean Field Games systems, and we study the convergence of this procedure to a solution of the problem. We also introduce suitable discretizations to numerically solve both stationary and evolutive problems. We show the convergence of the policy iteration method for the discrete problem and we study the performance of the proposed algorithm on some examples in dimension one and two.

scholarly journals Finite-State Contract Theory with a Principal and a Field of Agents

2020 ◽  
Author(s):  
René Carmona ◽  
Peiqi Wang
Keyword(s):  
Optimal Control ◽  
Stochastic Models ◽  
Contract Theory ◽  
Large Population ◽  
Mean Field ◽  
Theory Model ◽  
Mean Field Games ◽  
Weak Formulation ◽  
Finite State ◽  
Set Up

We use the recently developed probabilistic analysis of mean field games with finitely many states in the weak formulation to set up a principal/agent contract theory model where the principal faces a large population of agents interacting in a mean field manner. We reduce the problem to the optimal control of dynamics of the McKean-Vlasov type, and we solve this problem explicitly for a class of models with concave rewards. The paper concludes with a numerical example demonstrating the power of the results when applied to an example of epidemic containment. This paper was accepted by Baris Ata, stochastic models and simulation.

On closed-loop equilibrium strategies for mean-field stochastic linear quadratic problems

2020 ◽  
Vol 26 ◽  
pp. 41
Author(s):  
Tianxiao Wang
Keyword(s):  
Optimal Control ◽  
Closed Loop ◽  
Optimal Control Problems ◽  
Mean Field ◽  
Open Loop ◽  
Time Inconsistency ◽  
Control Problems ◽  
Linear Quadratic ◽  
Linear Quadratic Optimal Control ◽  
Equilibrium Strategies

This article is concerned with linear quadratic optimal control problems of mean-field stochastic differential equations (MF-SDE) with deterministic coefficients. To treat the time inconsistency of the optimal control problems, linear closed-loop equilibrium strategies are introduced and characterized by variational approach. Our developed methodology drops the delicate convergence procedures in Yong [Trans. Amer. Math. Soc. 369 (2017) 5467–5523]. When the MF-SDE reduces to SDE, our Riccati system coincides with the analogue in Yong [Trans. Amer. Math. Soc. 369 (2017) 5467–5523]. However, these two systems are in general different from each other due to the conditional mean-field terms in the MF-SDE. Eventually, the comparisons with pre-committed optimal strategies, open-loop equilibrium strategies are given in details.

Sign in / Sign up

Export Citation Format

Share Document