Dynamic Mean-Variance Asset Allocation in General Incomplete Markets

Abstract. This paper studies dynamic mean-variance (MV) asset allocation problems in general incomplete markets. Besides the conventional MV objective on a portfolio’s terminal wealth, our framework can accommodate running MV objectives with general (nonexponential) discounting factors and, in general, any time-dependent preferences. We attempt the problem with a game-theoretic framework and decompose the equilibrium control policies into two parts: the first part is a myopic strategy characterized by a linear Volterra integral equation of the second kind, and the second part reveals the hedging demand governed by a system of nonlocal backward stochastic differential equations. We manage to establish the global well-posedness of the solutions to the two aforementioned equations in tailored Bananch spaces by the fixed-point theorem. It allows us to devise a numerical scheme for solving for the equilibrium control policy with a guarantee and to conclude that the dynamic (equilibrium) mean-variance policy in general settings is well-defined. Our probabilistic approach allows us to consider a broad range of stochastic factor models, such as the Chan–Karolyi–Longstaff–Sanders (CKLS) model. For this model, we verify all technical assumptions and provide a sound numerical scheme. Numerical examples are provided to illustrate our framework.