Continuous-time portfolio selection with liability: Mean-variance model and stochastic LQ approach

In this paper we formulate a continuous-time mean-variance portfolio selection model with multiple risky assets and one liability in an incomplete market. The risky assets' prices are governed by geometric Brownian motions while the liability evolves according to a Brownian motion with drift. The correlations between the risky assets and the liability are considered. The objective is to maximize the expected terminal wealth while minimizing the variance of the terminal wealth. We derive explicitly the optimal dynamic strategy and the mean-variance efficient frontier in closed forms by using the general stochastic linear-quadratic (LQ) control technique. Several special cases are discussed and a numerical example is also given.