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
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.
Place, publisher, year, edition, pages
2009. Vol. 58, no 1, 76-82 p.
Adjoint equations; Martingale measures; Maximum principle; Relaxed control; Singular control
Probability Theory and Statistics
IdentifiersURN: urn:nbn:se:kth:diva-13459DOI: 10.1016/j.sysconle.2008.08.003ISI: 000262755200011ScopusID: 2-s2.0-57249094306OAI: oai:DiVA.org:kth-13459DiVA: diva2:325412
QC 201006182010-06-182010-06-182011-03-16Bibliographically approved