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The relaxed general maximum principle for singular optimal control of diffusions
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
2009 (Engelska)Ingår i: Systems & control letters (Print), ISSN 0167-6911, E-ISSN 1872-7956, ISSN 01676911, Vol. 58, nr 1, s. 76-82Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

In this paper we study optimality in stochastic control problems where the state process is a stochastic differential equation (SDE) and the control variable has two components, the first being absolutely continuous and the second singular. A control is defined as a solution to the corresponding martingale problem. To obtain existence of an optimal control Haussmann and Suo [U.G. Haussmann, W. Suo, Singular optimal stochastic controls I: Existence, SIAM J. Control Optim. 33 (3) (1995) 916-936] relaxed the martingale problem by extending the absolutely continuous control to the space of probability measures on the control set. Bahlali et al. [S. Bahlali, B. Djehiche, B. Mezerdi, The relaxed stochastic maximum principle in singular optimal control of diffusions, SIAM J. Control Optim. 46 (2) (2007) 427-444] established a maximum principle for relaxed singular control problems with uncontrolled diffusion coefficient. The main goal of this paper is to extend their results to the case where the control enters the diffusion coefficient. The proof is based on necessary conditions for near optimality of a sequence of ordinary controls which approximate the optimal relaxed control. The necessary conditions for near optimality are obtained by Ekeland's variational principle and the general maximum principle for (strict) singular control problems obtained in Bahlali and Mezerdi [S. Bahlali, B. Mezerdi, A general stochastic maximum principle for singular control problems, Electron J. Probab. 10 (2005) 988-1004. Paper no 30]. © 2008 Elsevier B.V. All rights reserved.

Ort, förlag, år, upplaga, sidor
2009. Vol. 58, nr 1, s. 76-82
Nyckelord [en]
Adjoint equations; Martingale measures; Maximum principle; Relaxed control; Singular control
Nationell ämneskategori
Sannolikhetsteori och statistik
Identifikatorer
URN: urn:nbn:se:kth:diva-13459DOI: 10.1016/j.sysconle.2008.08.003ISI: 000262755200011Scopus ID: 2-s2.0-57249094306OAI: oai:DiVA.org:kth-13459DiVA, id: diva2:325412
Anmärkning
QC 20100618Tillgänglig från: 2010-06-18 Skapad: 2010-06-18 Senast uppdaterad: 2017-12-12Bibliografiskt granskad
Ingår i avhandling
1. 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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