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Mean Field Games for Jump Non-Linear Markov Process
Linnaeus University, Faculty of Technology, Department of Mathematics.
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The mean-field game theory is the study of strategic decision making in very large populations of weakly interacting individuals. Mean-field games have been an active area of research in the last decade due to its increased significance in many scientific fields. The foundations of mean-field theory go back to the theory of statistical and quantum physics. One may describe mean-field games as a type of stochastic differential game for which the interaction between the players is of mean-field type, i.e the players are coupled via their empirical measure. It was proposed by Larsy and Lions and independently by Huang, Malhame, and Caines. Since then, the mean-field games have become a rapidly growing area of research and has been studied by many researchers. However, most of these studies were dedicated to diffusion-type games. The main purpose of this thesis is to extend the theory of mean-field games to jump case in both discrete and continuous state space. Jump processes are a very important tool in many areas of applications. Specifically, when modeling abrupt events appearing in real life. For instance, financial modeling (option pricing and risk management), networks (electricity and Banks) and statistics (for modeling and analyzing spatial data). The thesis consists of two papers and one technical report which will be submitted soon:

In the first publication, we study the mean-field game in a finite state space where the dynamics of the indistinguishable agents is governed by a controlled continuous time Markov chain. We have studied the control problem for a representative agent in the linear quadratic setting. A dynamic programming approach has been used to drive the Hamilton Jacobi Bellman equation, consequently, the optimal strategy has been achieved. The main result is to show that the individual optimal strategies for the mean-field game system represent 1/N-Nash equilibrium for the approximating system of N agents.

As a second article, we generalize the previous results to agents driven by a non-linear pure jump Markov processes in Euclidean space. Mathematically, this means working with linear operators in Banach spaces adapted to the integro-differential operators of jump type and with non-linear partial differential equations instead of working with linear transformations in Euclidean spaces as in the first work. As a by-product, a generalization for the Koopman operator has been presented. In this setting, we studied the control problem in a more general sense, i.e. the cost function is not necessarily of linear quadratic form. We showed that the resulting unique optimal control is of Lipschitz type. Furthermore, a fixed point argument is presented in order to construct the approximate Nash Equilibrium. In addition, we show that the rate of convergence will be of special order as a result of utilizing a non-linear pure jump Markov process.

In a third paper, we develop our approach to treat a more realistic case from a modelling perspective. In this step, we assume that all players are subject to an additional common noise of Brownian type. We especially study the well-posedness and the regularity for a jump version of the stochastic kinetic equation. Finally, we show that the solution of the master equation, which is a type of second order partial differential equation in the space of probability measures, provides an approximate Nash Equilibrium. This paper, unfortunately, has not been completely finished and it is still in preprint form. Hence, we have decided not to enclose it in the thesis. However, an outlook about the paper will be included.

Place, publisher, year, edition, pages
Växjö: Linnaeus University Press, 2016. , 118 p.
Series
Linnaeus University Dissertations, 260/2016
Keyword [en]
Mean-field games, Optimal Control, Non-linear Markov Processes
National Category
Probability Theory and Statistics
Research subject
Natural Science, Mathematics
Identifiers
URN: urn:nbn:se:lnu:diva-55852ISBN: 978-91-88357-30-4OAI: oai:DiVA.org:lnu-55852DiVA: diva2:956697
Public defence
2016-09-16, D1136V, Hus D, Växjö, 10:15 (English)
Opponent
Supervisors
Available from: 2016-08-31 Created: 2016-08-31 Last updated: 2016-09-13Bibliographically approved
List of papers
1. An Approximate Nash Equilibriumfor Pure Jump Markov Games of Mean-field-type on Continuous State Space
Open this publication in new window or tab >>An Approximate Nash Equilibriumfor Pure Jump Markov Games of Mean-field-type on Continuous State Space
(English)Manuscript (preprint) (Other academic)
Abstract
Keyword
Mean-field games-Non-linear Markov Processes, Optimal Control
National Category
Probability Theory and Statistics
Research subject
Mathematics, Mathematics
Identifiers
urn:nbn:se:lnu:diva-55851 (URN)
External cooperation:
Projects
Sweden
Available from: 2016-08-31 Created: 2016-08-31 Last updated: 2016-09-07
2. An Epsilon Nash Equilibrium For Non-Linear Markov Games of Mean-Field-Type on Finite Spaces
Open this publication in new window or tab >>An Epsilon Nash Equilibrium For Non-Linear Markov Games of Mean-Field-Type on Finite Spaces
2014 (English)In: Communications on Stochastic Analysis, ISSN 0973-9599, Vol. 8, no 4, 449-468 p.Article in journal (Refereed) Published
Abstract [en]

We investigate mean field games from the point of view of a large number of indistinguishable players which eventually converges to in- finity. The players are weakly coupled via their empirical measure. The dynamics of the individual players is governed by pure jump type propagators over a finite space. Investigations are conducted in the framework of non-linear Markov processes. We show that the individual optimal strategy results from a consistent coupling of an optimal control problem with a forward non-autonomous dynamics. In the limit as the number N of players goes to infinity this leads to a jump-type analog of the well-known non-linear McKean-Vlasov dynamics. The case where one player has an individual preference different from the ones of the remaining players is also covered. The two results combined reveal a 1 N -Nash Equilibrium for the approximating system of N players.

National Category
Mathematics Probability Theory and Statistics
Identifiers
urn:nbn:se:lnu:diva-40159 (URN)
Available from: 2015-02-15 Created: 2015-02-15 Last updated: 2016-08-31Bibliographically approved

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