Discrete-time mean variance optimal control of linear systems with Markovian jumps and multiplicative noise

Abstract

In this article, we consider the stochastic optimal control problem of discrete-time linear systems subject to Markov jumps and multiplicative noise under three kinds of performance criterions related to the final value of the expectation and variance of the output. In the first problem it is desired to minimise the final variance of the output subject to a restriction on its final expectation, in the second one it is desired to maximise the final expectation of the output subject to a restriction on its final variance, and in the third one it is considered a performance criterion composed by a linear combination of the final variance and expectation of the output of the system. We present explicit sufficient conditions for the existence of an optimal control strategy for these problems, generalising previous results in the literature. We conclude this article presenting a numerical example of an asset liabilities management model for pension funds with regime switching.

Keywords:

• mean variance control
• Markov jump systems
• multiplicative noise
• stochastic optimal control

Acknowledgements

O.L.V. Costa received financial support from CNPq (Brazilian National Research Council), Grant 304866/03-2, and FAPESP (Research Council of the State of São Paulo), Grant 03/06736-7.