| A financial market consists of numerous securities. A typical investor invests his initial capital in different securities and carry out continuous trading of them to increase the final wealth of his portfolio. Thus from the point of view of an investor it is important to know which choice of portfolio would result in the best return. This problem is termed as optimal portfolio selection problem. This is, in principle, a dynamic optimization problem. But on a general level, investment managers and academic economists have long been aware of the necessity of taking returns as well as risk into account: ''all the eggs should not be placed in the same basket". There are many different approaches to solve this problem. The relevance of the approaches depend on the investor's interest. An investor may want to optimize the final mean wealth of the portfolio by keeping the risk level (the variance of the return) fixed. This is known as the mean-variance approach. The mean-variance approach was proposed by Markowitz (Harry M. Markowitz was one of the three recipients of Nobel Prize in Economics in 1990 for developing the theory of portfolio choice) for portfolio construction in a single period. This provides an operational theory for portfolio selection under uncertainty. Subsequently the theory is extended for multi-period trading and finally for continuous time trading and it evolved into a foundation for further research in financial economics.
nstead of optimizing the expected terminal wealth, sometimes the investor may have a preferred utility function and may like to optimize the expected utility of terminal wealth. But in this approach one cannot take care of the dispersion and more importantly, the choice of utility function is subject of investor's preference. There is another popular approach where the investor optimizes the risk sensitive criterion of final wealth. This approach enables the investor to specify his aversion of risk (parameterized by the asymptotic variance) in advance.
We address a portfolio optimization problem in a market that is modeled using a semi-Markov modulated geometric Brownian motion. In this talk risk-sensitive portfolio optimization on finite and infinite time horizon are considered. We use a probabilistic method to establish the existence and uniqueness of classical solution of the HJB equation for finite horizon problem. A numerical procedure is also developed to compute the optimal expected terminal utility for finite horizon problem. We show that the optimal portfolio selection remains the same for both finite and infinite horizon cases. The computation of the optimal growth rate for the infinite horizon case is more involved since this involves the large deviation principle for semi-Markov processes which does not seem to be available in the literature. In the special case involving Markovian switching, we obtained the optimal growth rate in terms of a maximal eigenvalue of an appropriate matrix, using Perron-Frobenius theorem. |