Markowitz's Mean-Variance Portfolio Selection with Regime Switching: A Continuous-Time Model

TLDR

Regime switching causes market parameters such as interest, appreciation, and volatility rates to change among a finite set of states, and this switching is assumed independent of the underlying Brownian motion. The paper proposes and analyzes a continuous‑time Markowitz mean‑variance portfolio selection model for a market with one bank account and multiple stocks under regime switching. Using a Markov‑chain–modulated diffusion framework and stochastic linear‑quadratic control, the authors solve two systems of linear ODEs to obtain explicit mean‑variance efficient portfolios, efficient frontiers, and address minimum‑variance portfolios and a mutual‑fund theorem. The resulting market is incomplete, yet the model yields closed‑form efficient portfolios and frontiers that differ markedly from the no‑regime‑switching case, except when the interest rate is deterministic, in which case the results resemble the classic case.

Abstract

A continuous-time version of the Markowitz mean-variance portfolio selection model is proposed and analyzed for a market consisting of one bank account and multiple stocks. The market parameters, including the bank interest rate and the appreciation and volatility rates of the stocks, depend on the market mode that switches among a finite number of states. The random regime switching is assumed to be independent of the underlying Brownian motion. This essentially renders the underlying market incomplete. A Markov chain modulated diffusion formulation is employed to model the problem. Using techniques of stochastic linear-quadratic control, mean-variance efficient portfolios and efficient frontiers are derived explicitly in closed forms, based on solutions of two systems of linear ordinary differential equations. Related issues such as a minimum-variance portfolio and a mutual fund theorem are also addressed. All the results are markedly different from those for the case when there is no regime switching. An interesting observation is, however, that if the interest rate is deterministic, then the results exhibit (rather unexpected) similarity to their no-regime-switching counterparts, even if the stock appreciation and volatility rates are Markov-modulated.