Nash equilibrium strategy for a multi-period mean–variance portfolio selection problem with regime switching

This paper additionally incorporates the random decision behaviors of the decision-maker, in addiction to the randomness in the risky assets, into the decision-making process. Our work to this point is to investigate a multi-period mean–variance portfolio optimization under the assumption that the risk aversion is changeable according to the macroeconomic market state as the returns of the risky assets do. It is well known that the Markowitz's mean–variance portfolio selection problem is time-inconsistent, especially when the risk aversion is assumed to be dynamically changeable. Within a game theoretic framework, we derived the (subgame perfect Nash) equilibrium strategy and equilibrium value function in closed-form. We identify some interesting properties of the equilibrium investment strategy, the equilibrium value function, the terminal variance and the efficient frontier under the equilibrium strategy through numerical sensitivity analysis.