, Volume 141, Issue 3, pp 677-700
Date: 21 Jan 2009

General Maximum Principles for Partially Observed Risk-Sensitive Optimal Control Problems and Applications to Finance

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Abstract

This paper is concerned with partially observed risk-sensitive optimal control problems. Combining Girsanov’s theorem with a standard spike variational technique, we obtain some general maximum principles for the aforementioned problems. One of the distinctive differences between our results and the standard risk-neutral case is that the adjoint equations and variational inequalities strongly depend on a risk-sensitive parameter γ. Two examples are given to illustrate the applications of the theoretical results obtained in this paper. As a natural deduction, a general maximum principle is also obtained for a fully observed risk-sensitive case. At last, this result is applied to study a risk-sensitive optimal portfolio problem. An explicit optimal investment strategy and a cost functional are obtained. A numerical simulation result shows the influence of a risk-sensitive parameter on an optimal investment proportion; this coincides with its economic meaning and theoretical results.

Communicated by F. Zirilli.
This work was partially supported by the National Natural Science Foundation (10671112), the National Basic Research Program of China (973 Program, No. 2007CB814904), the Natural Science Foundation of Shandong Province (Z2006A01) and the Doctoral Fund of the Education Ministry of China.