Open-loop equilibrium strategy for mean-variance portfolio selection: A log-return model

This paper investigates a continuous-time mean-variance portfolio selection problem based on a log-return model. The financial market is composed of one risk-free asset and multiple risky assets whose prices are modelled by geometric Brownian motions. We derive a sufficient condition for open-loop equilibrium strategies via forward backward stochastic differential equations (FBSDEs). An equilibrium strategy is derived by solving the system. To illustrate our result, we consider a special case where the interest rate process is described by the Vasicek model. In this case, we also derive the closed-loop equilibrium strategy through the dynamic programming approach.