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Keywords:

  • portfolio optimization;
  • hidden Markov chain;
  • sufficient statistics;
  • dynamic programming

Abstract

We consider a utility-based portfolio selection problem, where the parameters change according to a Markovian market that cannot be observed perfectly. The market consists of a riskless and many risky assets whose returns depend on the state of the unobserved market process. The states of the market describe the prevailing economic, financial, social, political or other conditions that affect the deterministic and probabilistic parameters of the model. However, investment decisions are based on the information obtained by the investors. This constitutes our observation process. Therefore, there is a Markovian market process whose states are unobserved, and a separate observation process whose states are observed by the investors who use this information to determine their portfolios. There is, of course, a probabilistic relation between the two processes. The market process is a hidden Markov chain and we use sufficient statistics to represent the state of our financial system. The problem is solved using the dynamic programming approach to obtain an explicit characterization of the optimal policy and the value function. In particular, the return-risk frontiers of the terminal wealth are shown to have linear forms. Copyright © 2011 John Wiley & Sons, Ltd.