Optimal control with constrained total variance for Markov jump linear systems with multiplicative noises

ABSTRACT

We consider in this paper the stochastic optimal control problem of discrete-time linear systems subject to Markov jumps and multiplicative noises (MJLS-mn for short). Our objective is to present an optimal policy for the problem of maximising the system's total expected output over a finite-time horizon while restricting the weighted sum of its variance to a pre-specified upper-bound value. We obtain explicit conditions for the existence of an optimal control law for this problem as well as an algorithm for obtaining it, extending previous results in the literature. The paper is concluded by applying our results to a portfolio selection problem subject to regime switching.

KEYWORDS:

• Markov jump systems
• variance constrain
• multiplicative noises
• multi-period optimal control
• Riccati equations
• mean-variance control
• optimal portfolio

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

F. Barbieri was supported by Fundação CAPES [USP-scholarship] (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior); O.L.V. Costa was supported in part by the Brazilian National Research Council – CNPq [grant number 304091/2014-6]; São Paulo Research Foundation – FAPESP/Shell through the Research Center for Gas Innovation[grant number 2014/50279-4], and project INCT (National Institute of Science and Technology) [grant number CNPq 465755/2014-3, FAPESP 2014/50851-0].

Notes on contributors

Fabio Barbieri

Fabio Barbieri was born in 1978 in São Paulo, Brazil, He obtained his B.Sc degree in Aeronautical Engineering from ITA, Brazil, in 2003, and the M.Sc. degree in Electrical Engineering from the Polytechnic School of the University of São Paulo, Brazil, in 2016. He is currently working toward the Ph.D degree at the Department of Telecommunications and Control Engineering of the Polytechnic School of the University of São Paulo, Brazil. His research focuses mainly on stochastic control and artificial intelligence.

Oswaldo L. V. Costa

Oswaldo L. V. Costa was born in 1959 in Rio de Janeiro, RJ, Brazil. He obtained his B.Sc and M.Sc degrees both in Electrical Engineering from the Catholic University of Rio de Janeiro, Brazil, in 1981 and 1983 respectively, and the Ph.D degree in Electrical Engineering from the Imperial College of Science and Technology in London in 1987. He is presently a Professor in the Control Group of the Department of Telecommunications and Control Engineering of the Polytechnic School of the University of São Paulo, Brazil. His research interests include stochastic control, optimal control, and jump systems.