Maximum Principle for Risk-Sensitive Stochastic Optimal Control Problem and Applications to Finance

Abstract

This article is concerned with a risk-sensitive stochastic optimal control problem motivated by a kind of optimal portfolio choice problem in the financial market. The maximum principle for this kind of problem is obtained, which is similar in form to its risk-neutral counterpart. But the adjoint equations and maximum condition heavily depend on the risk-sensitive parameter. This result is used to solve a kind of optimal portfolio choice problem and the optimal portfolio choice strategy is obtained. Computational results and figures explicitly illustrate the optimal solution and the sensitivity to the volatility rate parameter.

Keywords:

• Maximum principle
• Ornstein-Uhlenbeck model
• Riccati equation
• Risk-sensitive control
• Stochastic optimal control

Mathematics Subject Classification:

• Primary 93E20, 60H10
• Secondary 49K45, 60H30

Acknowledgments

Research of J. Shi is supported by the China Postdoctoral Science Foundation Funded Project (No. 20100481278), Postdoctoral Innovation Foundation Funded Project of Shandong Province (No. 201002026), National Natural Science Foundations of China (No. 11126209) and Shandong Province (No. ZR2011AQ012), and Independent Innovation Foundation of Shandong University (IIFSDU, No. 2010TS060). Research of Z. Wu is supported by National Natural Sciences Foundations of China (No. 11921101, 61174092) and National Natural Sciences Fund for Distinguished Young Scholars of China (No. 11125102).