Mean-Variance Portfolio Selection in a Markovian Regime-Switching Market When the Uncertain Time Horizon is a Stopping Time of Market State Filtration: A Multi-period Model

This paper considers the Markowitz's mean-variance portfolio selection model in a multi-period setting with regime switching and uncertain time horizon. The returns of the assets depend on the state of the market modulated by a discrete-time Markov chain with a finite state space. The exit time from the market is a stopping time with respect to the market state filtration. So, the definitive decision to exit the market at any time depends only on the market states up to that time. The original problem with uncertain exit time is reformulated as a problem with certain exit time. The Lagrange duality method and the dynamic programming approach are used to derive explicit closed-form expressions for the efficient investment strategy and the mean-variance efficient frontier. Toward this objective, a market path-dependent value function method is introduced and it is shown that optimal portfolios are market path dependent. The two cases where the exit time is the first hitting time to a specified subset of the market state space and the market state contains a bankruptcy state are investigated separately. Moreover, it is shown that some results in the existing literature can be obtained as special cases of the results in this paper. Finally, some numerical examples are presented to illustrate the results.