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A Stochastic Maximum Principle for Markov Chains of Mean-Field Type
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
2018 (English)In: Games, ISSN 2073-4336, E-ISSN 2073-4336, Vol. 9, no 4, article id 84Article in journal (Refereed) Published
Abstract [en]

We derive sufficient and necessary optimality conditions in terms of a stochastic maximum principle (SMP) for controls associated with cost functionals of mean-field type, under dynamics driven by a class of Markov chains of mean-field type which are pure jump processes obtained as solutions of a well-posed martingale problem. As an illustration, we apply the result to generic examples of control problems as well as some applications. 

Place, publisher, year, edition, pages
MDPI AG , 2018. Vol. 9, no 4, article id 84
Keywords [en]
mean-field; nonlinear Markov chain; backward SDEs; optimal control; stochastic maximum principle
National Category
Probability Theory and Statistics
Research subject
Applied and Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-240109DOI: 10.3390/g9040084Scopus ID: 2-s2.0-85056273906OAI: oai:DiVA.org:kth-240109DiVA, id: diva2:1269916
Funder
Swedish Research Council, 2016-04086
Note

QC 20181212

Available from: 2018-12-11 Created: 2018-12-11 Last updated: 2018-12-12Bibliographically approved
In thesis
1. Topics in Mean-Field Control and Games for Pure Jump Processes
Open this publication in new window or tab >>Topics in Mean-Field Control and Games for Pure Jump Processes
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis is the collection of four papers addressing topics in stochastic optimal control, zero-sum games, backward stochastic differential equations, Pontryagin stochastic maximum principle and relaxed stochastic optimal control.

In the first two papers, we establish existence of Markov chains of mean-field type, with countable state space and unbounded jump intensities. We further show existence of nearly-optimal controls and, using a Markov chain backward SDE approach, we derive conditions for existence of an optimal control and a saddle-point for a zero-sum differential game associated with risk-neutral and risk-sensitive payoff functionals of mean-field type, under dynamics driven by Markov chains of mean-field type. Our formulation of the control problems is of weak-type, where the dynamics are given in terms of a family of probability measures, under which the coordinate process is a pure jump process with controlled jump intensities.

In the third paper, we characterize the optimal controls obtained in the first pa-per by deriving sufficient and necessary optimality conditions in terms of a stochastic maximum principle (SMP). Finally, within a completely different setup, in the fourth paper we establish existence of an optimal stochastic relaxed control for stochastic differential equations driven by a G-Brownian motion.

Place, publisher, year, edition, pages
KTH Royal Institute of Technology, 2018. p. 16
Series
TRITA-SCI-FOU ; 2018:55
National Category
Probability Theory and Statistics
Research subject
Applied and Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-240113 (URN)978-91-7873-061-2 (ISBN)
Public defence
2019-02-01, Kollegiesalen, Brinellvägen 8, Kungliga Tekniska högskolan, Stockholm, 10:00 (English)
Opponent
Supervisors
Funder
Swedish Research Council, 2016-04086
Note

QC 20181212

Available from: 2018-12-12 Created: 2018-12-12 Last updated: 2018-12-13Bibliographically approved

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