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Topics in Mean-Field Control and Games for Pure Jump Processes
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
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: urn:nbn:se:kth:diva-240113ISBN: 978-91-7873-061-2 (print)OAI: oai:DiVA.org:kth-240113DiVA, id: diva2:1270125
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
List of papers
1. Optimal Control and Zero-Sum Games for Markov Chains of Mean-Field Type
Open this publication in new window or tab >>Optimal Control and Zero-Sum Games for Markov Chains of Mean-Field Type
2018 (English)In: Mathematical Control and Related Fields, ISSN 2156-8472, E-ISSN 2156-8499Article in journal (Refereed) Accepted
Abstract [en]

We establish existence of Markov chains of mean-field type with unbounded jump intensities by means of a fixed point argument using the Total Variation distance. We further show existence of nearly-optimal controls and, using a Markov chain backward SDE approach, we suggest conditions for existence of an optimal control and a saddle-point for respectively a control problem and a zero-sum differential game associated with payoff functionals of mean-field type, under dynamics driven by such Markov chains of mean-field type.

Keywords
Mean-field, nonlinear Markov chain, backward SDEs, optimal control, zero-sum game, saddle point, stochastic maximum principle, thinning
National Category
Probability Theory and Statistics
Research subject
Applied and Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-240111 (URN)
Funder
Swedish Research Council, 2016-04086
Note

QC 20181212

Available from: 2018-12-11 Created: 2018-12-11 Last updated: 2018-12-12Bibliographically approved
2. Mean-Field Risk Sensitive Control and Zero-Sum Games for Markov Chains
Open this publication in new window or tab >>Mean-Field Risk Sensitive Control and Zero-Sum Games for Markov Chains
2018 (English)Manuscript (preprint) (Other academic)
Abstract [en]

We establish existence of controlled Markov chain of mean-field type with unbounded jump intensities by means of a fixed point argument using the Wasserstein distance. Using a Markov chain entropic backward SDE approach, we further suggest conditions for existence of an optimal control and a saddle-point for respectively a control problem and a zero-sum differential game associated with risk sensitive payoff functionals of mean-field type.

Keywords
mean-field, nonlinear Markov chain, backward SDE, entropy, optimal control, risk sensitive, zero-sum game, saddle-point.
National Category
Probability Theory and Statistics
Research subject
Applied and Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-240112 (URN)
Funder
Swedish Research Council, 2016-04086
Note

QC 20181212

Available from: 2018-12-11 Created: 2018-12-11 Last updated: 2018-12-17Bibliographically approved
3. A Stochastic Maximum Principle for Markov Chains of Mean-Field Type
Open this publication in new window or tab >>A Stochastic Maximum Principle for Markov Chains of Mean-Field Type
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
Keywords
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:nbn:se:kth:diva-240109 (URN)10.3390/g9040084 (DOI)2-s2.0-85056273906 (Scopus ID)
Funder
Swedish Research Council, 2016-04086
Note

QC 20181212

Available from: 2018-12-11 Created: 2018-12-11 Last updated: 2018-12-12Bibliographically approved
4. On relaxed stochastic optimal control for stochastic differential equations driven by G-Brownian motion
Open this publication in new window or tab >>On relaxed stochastic optimal control for stochastic differential equations driven by G-Brownian motion
2018 (English)In: Latin American Journal of Probability and Mathematical Statistics, ISSN 1980-0436, E-ISSN 1980-0436, Vol. 15, no 1, p. 201-212Article in journal (Refereed) Published
Abstract [en]

In the G-framework, we establish existence of an optimal stochastic relaxed control for stochastic differential equations driven by a G-Brownian motion.

Place, publisher, year, edition, pages
Instituto Nacional de Matematica Pura e Aplicada, 2018
Keywords
Capacity, G-Brownian motion, G-chattering lemma, Relaxed optimal control, Sublinear expectation
National Category
Probability Theory and Statistics
Research subject
Applied and Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-238436 (URN)10.30757/ALEA.v15-09 (DOI)2-s2.0-85052941773 (Scopus ID)
Note

QC 20181031

Available from: 2018-10-31 Created: 2018-10-31 Last updated: 2018-12-14Bibliographically approved

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