The maximum principles for partially observed risk-sensitive optimal controls of Markov regime-switching jump-diffusion system

ABSTRACT

This paper studies partially observed risk-sensitive optimal control problems with correlated noises between the system and the observation. It is assumed that the state process is governed by a continuous-time Markov regime-switching jump-diffusion process and the cost functional is of an exponential-of-integral type. By virtue of a classical spike variational approach, we obtain two general maximum principles for the aforementioned problems. Moreover, under certain convexity assumptions on both the control domain and the Hamiltonian, we give a sufficient condition for the optimality. For illustration, a linear-quadratic risk-sensitive control problem is proposed and solved using the main results. As a natural deduction, a fully observed risk-sensitive maximum principle is also obtained and applied to study a risk-sensitive portfolio optimization problem. Closed-form expressions for both the optimal portfolio and the corresponding optimal cost functional are obtained.

KEYWORDS:

• Partial information
• risk-sensitive control
• regime-switching
• jump-diffusion
• stochastic maximum principle

MATHEMATICS SUBJECT CLASSIFICATION:

• 93E20
• 91G80

Acknowledgments

The authors would like to thank the associate editor and the referee for their careful reading of the paper and helpful suggestions.

Additional information

Funding

The project on which this publication is based has been carried out with funding provided by the Alexander von Humboldt Foundation, under the programme financed by the German Federal Ministry of Education and Research entitled German Research Chair No. 01DG15010; the European Union’s Seventh Framework Programme for research, technological development, and demonstration under grant agreement No. 318984-RARE; the National Natural Science Foundation of China under grant No. 11571189 and the China Postdoctoral Science Foundation under grant No. 2017M612787.