CHINESE JOURNAL OF APPLIED PROBABILITY AND STATIST 2012, 28(5) 457-470 DOI:      ISSN: 1001-4268 CN: 31-1256

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Optimal Control for Utility Portfolio Selection with Liability

Chang Hao, Rong Ximin

Department of Mathematics, Tianjin Polytechnic University, School of Management, Tianjin University, School of Science, Tianjin University

Abstract��

In this paper we use stochastic optimal
control theory to investigate a dynamic portfolio selection problem
with liability process, in which the liability process is assumed to
be a geometric Brownian motion and completely correlated with stock
prices. We apply dynamic programming principle to obtain
Hamilton-Jacobi-Bellman (HJB) equations for the value function and
systematically study the optimal investment strategies for power
utility, exponential utility and logarithm utility. Firstly, the
explicit expressions of the optimal portfolios for power utility and
exponential utility are obtained by applying variable change
technique to solve corresponding HJB equations. Secondly, we apply
Legendre transform and dual approach to derive the optimal portfolio
for logarithm utility. Finally, numerical examples are given to
illustrate the results obtained and analyze the effects of the
market parameters on the optimal portfolios.

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