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Contributions to the Stochastic Maximum Principle
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
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: urn:nbn:se:kth:diva-11301ISBN: 978-91-7415-436-8 (print)OAI: oai:DiVA.org:kth-11301DiVA: diva2:272710
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
List of papers
1. A maximum principle for SDEs of mean-field type
Open this publication in new window or tab >>A maximum principle for SDEs of mean-field type
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.

National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-13441 (URN)10.1007/s00245-010-9123-8 (DOI)000288507800002 ()2-s2.0-79958262462 (Scopus ID)
Note
QC 20110411 uppdaterad från submitted till published 20110411Available from: 2010-06-18 Created: 2010-06-18 Last updated: 2017-12-12Bibliographically approved
2. 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
3. A mixed relaxed singular maximum principle for linear SDEs with random coefficients
Open this publication in new window or tab >>A mixed relaxed singular maximum principle for linear SDEs with random coefficients
(English)Article in journal (Refereed) Submitted
Abstract [en]

We study singular stochastic control of a two dimensional stochastic differential equation, where the first component is linear with random and unbounded coefficients. We derive existence of an optimal relaxed control and necessary conditions for optimality in the form of a mixed relaxed-singular maximum principle in a global form. A motivating example is given in the form of an optimal investment and consumption problem with transaction costs, where we consider a portfolio with a continuum of bonds and where the portfolio weights are modeled as measure-valued processes on the set of times to maturity.

Keyword
Optimization and Control
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-13454 (URN)
Note
QS 2012Available from: 2010-06-18 Created: 2010-06-18 Last updated: 2012-03-26Bibliographically approved
4. The relaxed general maximum principle for singular optimal control of diffusions
Open this publication in new window or tab >>The relaxed general maximum principle for singular optimal control of diffusions
2009 (English)In: Systems & control letters (Print), ISSN 0167-6911, E-ISSN 1872-7956, ISSN 01676911, Vol. 58, no 1, 76-82 p.Article in journal (Refereed) 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.

Keyword
Adjoint equations; Martingale measures; Maximum principle; Relaxed control; Singular control
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-13459 (URN)10.1016/j.sysconle.2008.08.003 (DOI)000262755200011 ()2-s2.0-57249094306 (Scopus ID)
Note
QC 20100618Available from: 2010-06-18 Created: 2010-06-18 Last updated: 2017-12-12Bibliographically approved

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