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A mixed relaxed singular maximum principle for linear SDEs with random coefficients
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
(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 [en]
Optimization and Control
National Category
Probability Theory and Statistics
URN: urn:nbn:se:kth:diva-13454OAI: diva2:325399
QS 2012Available from: 2010-06-18 Created: 2010-06-18 Last updated: 2012-03-26Bibliographically 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.
Trita-MAT, ISSN 1401-2286 ; 09:12
National Category
Probability Theory and Statistics
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)
QC 20100618Available from: 2009-10-16 Created: 2009-10-16 Last updated: 2010-07-19Bibliographically approved

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