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A mean-variance optimization problem for discounted Markov decision processes. (English) Zbl 1253.90214

Summary: We consider a mean-variance optimization problem for Markov decision processes (MDPs) over the set of (deterministic stationary) policies. Different from the usual formulation in MDPs, we aim to obtain the mean-variance optimal policy that minimizes the variance over a set of all policies with a given expected reward. For continuous-time MDPs with the discounted criterion and finite-state and action spaces, we prove that the mean-variance optimization problem can be transformed to an equivalent discounted optimization problem using the conditional expectation and Markov properties. Then, we show that a mean-variance optimal policy and the efficient frontier can be obtained by policy iteration methods with a finite number of iterations. We also address related issues such as a mutual fund theorem and illustrate our results with an example.

MSC:

90C40 Markov and semi-Markov decision processes
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