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A maximum principle for SDEs of mean-field type
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.ORCID iD: 0000-0002-6608-0715
2011 (English)In: Applied mathematics and optimization, ISSN 0095-4616, E-ISSN 1432-0606, Vol. 63, no 3, 341-356 p.Article in journal (Refereed) Published
Abstract [en]

We study the optimal control of a stochastic differential equation (SDE) of mean-field type, where the coefficients are allowed to depend on some functional of the law as well as the state of the process. Moreover the cost functional is also of mean-field type, which makes the control problem time inconsistent in the sense that the Bellman optimality principle does not hold. Under the assumption of a convex action space a maximum principle of local form is derived, specifying the necessary conditions for optimality. These are also shown to be sufficient under additional assumptions. This maximum principle differs from the classical one, where the adjoint equation is a linear backward SDE, since here the adjoint equation turns out to be a linear mean-field backward SDE. As an illustration, we apply the result to the mean-variance portfolio selection problem.

Place, publisher, year, edition, pages
2011. Vol. 63, no 3, 341-356 p.
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:kth:diva-13441DOI: 10.1007/s00245-010-9123-8ISI: 000288507800002Scopus ID: 2-s2.0-79958262462OAI: oai:DiVA.org:kth-13441DiVA: diva2:325377
Note
QC 20110411 uppdaterad från submitted till published 20110411Available from: 2010-06-18 Created: 2010-06-18 Last updated: 2017-12-12Bibliographically approved
In thesis
1. Contributions to the Stochastic Maximum Principle
Open this publication in new window or tab >>Contributions to the Stochastic Maximum Principle
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of four papers treating the maximum principle for stochastic control problems.

In the first paper we study the optimal control of a class of stochastic differential equations (SDEs) of mean-field type, where the coefficients are allowed to depend on the law of the process. Moreover, the cost functional of the control problem may also depend on the law of the process. Necessary and sufficient conditions for optimality are derived in the form of a maximum principle, which is also applied to solve the mean-variance portfolio problem.

In the second paper, we study the problem of controlling a linear SDE where the coefficients are random and not necessarily bounded. We consider relaxed control processes, i.e. the control is defined as a process taking values in the space of probability measures on the control set. The main motivation is a bond portfolio optimization problem. The relaxed control processes are then interpreted as the portfolio weights corresponding to different maturity times of the bonds. We establish existence of an optimal control and necessary conditons for optimality in the form of a maximum principle, extended to include the family of relaxed controls.

The third paper generalizes the second one by adding a singular control process to the SDE. That is, the control is singular with respect to the Lebesgue measure and its influence on the state is thus not continuous in time. In terms of the portfolio problem, this allows us to consider two investment possibilities - bonds (with a continuum of maturities) and stocks - and incur transaction costs between the two accounts.

In the fourth paper we consider a general singular control problem. The absolutely continuous part of the control is relaxed in the classical way, i.e. the generator of the corresponding martingale problem is integrated with respect to a probability measure, guaranteeing the existence of an optimal control. This is shown to correspond to an SDE driven by a continuous orthogonal martingale measure. A maximum principle which describes necessary conditions for optimal relaxed singular control is derived.

Place, publisher, year, edition, pages
Stockholm: KTH, 2009. v, 15 p.
Series
Trita-MAT, ISSN 1401-2286 ; 09:12
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-11301 (URN)978-91-7415-436-8 (ISBN)
Public defence
2009-10-30, Sal F3, Lindstedtsvägen 26, KTH, Stockholm, 13:00 (English)
Opponent
Supervisors
Note
QC 20100618Available from: 2009-10-16 Created: 2009-10-16 Last updated: 2010-07-19Bibliographically approved

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