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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 219397, 12 pages
http://dx.doi.org/10.1155/2013/219397
Research Article

An Investment and Consumption Problem with CIR Interest Rate and Stochastic Volatility

1Department of Mathematics, Tianjin Polytechnic University, Binshui West Road 399, 300387 Tianjin, China
2School of Science, Tianjin University, Wei-jin Road 72, 300072 Tianjin, China

Received 6 March 2013; Accepted 13 May 2013

Academic Editor: Ryan Loxton

Copyright © 2013 Hao Chang and Xi-min Rong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We are concerned with an investment and consumption problem with stochastic interest rate and stochastic volatility, in which interest rate dynamic is described by the Cox-Ingersoll-Ross (CIR) model and the volatility of the stock is driven by Heston’s stochastic volatility model. We apply stochastic optimal control theory to obtain the Hamilton-Jacobi-Bellman (HJB) equation for the value function and choose power utility and logarithm utility for our analysis. By using separate variable approach and variable change technique, we obtain the closed-form expressions of the optimal investment and consumption strategy. A numerical example is given to illustrate our results and to analyze the effect of market parameters on the optimal investment and consumption strategies.