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A maximum principle for relaxed stochastic control of linear SDEs with application to bond portfolio optimization
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.ORCID-id: 0000-0002-6608-0715
2010 (Engelska)Ingår i: Mathematical Methods of Operations Research, ISSN 1432-2994, E-ISSN 1432-5217, Vol. 72, nr 2, s. 273-310Artikel i tidskrift (Refereegranskat) 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.

Ort, förlag, år, upplaga, sidor
2010. Vol. 72, nr 2, s. 273-310
Nyckelord [en]
Optimization and Control, Stochastic control, Relaxed control, Maximum principle, H-function, Bond portfolio
Nationell ämneskategori
Sannolikhetsteori och statistik
Identifikatorer
URN: urn:nbn:se:kth:diva-8006DOI: 10.1007/s00186-010-0320-7ISI: 000283255600005Scopus ID: 2-s2.0-78049527381OAI: oai:DiVA.org:kth-8006DiVA, id: diva2:13211
Anmärkning
QC 20100618 Ändrat från submitted till published 20110129Tillgänglig från: 2008-02-20 Skapad: 2008-02-20 Senast uppdaterad: 2017-12-14Bibliografiskt granskad
Ingår i avhandling
1. Necessary Optimality Conditions for Two Stochastic Control Problems
Öppna denna publikation i ny flik eller fönster >>Necessary Optimality Conditions for Two Stochastic Control Problems
2008 (Engelska)Licentiatavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Stockholm: KTH, 2008. s. vii
Serie
Trita-MAT, ISSN 1401-2286 ; 2008-MS-01
Nationell ämneskategori
Sannolikhetsteori och statistik
Identifikatorer
urn:nbn:se:kth:diva-4643 (URN)978-91-7178-887-0 (ISBN)
Presentation
2008-03-11, Sal E2, Lindstedtsv. 3, Stockholm, 15:15
Opponent
Handledare
Anmärkning
QC 20101102Tillgänglig från: 2008-02-20 Skapad: 2008-02-20 Senast uppdaterad: 2010-11-02Bibliografiskt granskad
2. Contributions to the Stochastic Maximum Principle
Öppna denna publikation i ny flik eller fönster >>Contributions to the Stochastic Maximum Principle
2009 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Stockholm: KTH, 2009. s. v, 15
Serie
Trita-MAT, ISSN 1401-2286 ; 09:12
Nationell ämneskategori
Sannolikhetsteori och statistik
Identifikatorer
urn:nbn:se:kth:diva-11301 (URN)978-91-7415-436-8 (ISBN)
Disputation
2009-10-30, Sal F3, Lindstedtsvägen 26, KTH, Stockholm, 13:00 (Engelska)
Opponent
Handledare
Anmärkning
QC 20100618Tillgänglig från: 2009-10-16 Skapad: 2009-10-16 Senast uppdaterad: 2010-07-19Bibliografiskt granskad

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