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Necessary Optimality Conditions for Two Stochastic Control Problems
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2008 (English)Licentiate thesis, comprehensive summary (Other scientific)
Abstract [en]

This thesis consists of two papers concerning necessary conditions in stochastic control problems. In the first paper, we study the problem of controlling a linear stochastic differential equation (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 conditions for optimality in the form of a maximum principle, extended to include the family of relaxed controls.

In the second paper we consider the so-called singular control problem where the control consists of two components, one absolutely continuous and one singular. The absolutely continuous part of the control is allowed to enter both the drift and diffusion coefficient. The absolutely continuous part 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 , 2008. , vii p.
Series
Trita-MAT, ISSN 1401-2286 ; 2008-MS-01
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:kth:diva-4643ISBN: 978-91-7178-887-0 (print)OAI: oai:DiVA.org:kth-4643DiVA: diva2:13213
Presentation
2008-03-11, Sal E2, Lindstedtsv. 3, Stockholm, 15:15
Opponent
Supervisors
Note
QC 20101102Available from: 2008-02-20 Created: 2008-02-20 Last updated: 2010-11-02Bibliographically approved
List of papers
1. A maximum principle for relaxed stochastic control of linear SDEs with application to bond portfolio optimization
Open this publication in new window or tab >>A maximum principle for relaxed stochastic control of linear SDEs with application to bond portfolio optimization
2010 (English)In: Mathematical Methods of Operations Research, ISSN 1432-2994, E-ISSN 1432-5217, Vol. 72, no 2, 273-310 p.Article in journal (Refereed) Published
Abstract [en]

We study relaxed stochastic control problems where the state equation is a one dimensional linear stochastic differential equation with random and unbounded coefficients. The two main results are existence of an optimal relaxed control and necessary conditions for optimality in the form of a relaxed maximum principle. The main motivation is an optimal bond portfolio problem in a market where there exists a continuum of bonds and the portfolio weights are modeled as measure-valued processes on the set of times to maturity.

Keyword
Optimization and Control, Stochastic control, Relaxed control, Maximum principle, H-function, Bond portfolio
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-8006 (URN)10.1007/s00186-010-0320-7 (DOI)000283255600005 ()2-s2.0-78049527381 (Scopus ID)
Note
QC 20100618 Ändrat från submitted till published 20110129Available from: 2008-02-20 Created: 2008-02-20 Last updated: 2017-12-14Bibliographically approved
2. The relaxed stochastic maximum principle in singular optimal control of diffusions with controlled diffusion coefficient
Open this publication in new window or tab >>The relaxed stochastic maximum principle in singular optimal control of diffusions with controlled diffusion coefficient
(English)Manuscript (Other academic)
Keyword
Singular control, Maximum principle, Adjoint equations, Relaxed control, Martingale measures
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-8007 (URN)
Note
QC 20101102Available from: 2008-02-20 Created: 2008-02-20 Last updated: 2010-11-02Bibliographically approved

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CiteExportLink to record
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Citation style
  • apa
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Output format
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