Time-consistent mean–variance asset–liability management with random coefficients

In this paper, we aim to find a time-consistent open-loop equilibrium strategy for the asset–liability management problem under mean–variance criterion. The financial market consists of a bank account and m stocks whose prices are modeled by geometric Brownian motions. The liability of the investor is uncontrollable and modeled by another geometric Brownian motion which is correlated to the stock prices. First, we provide a sufficient condition for the equilibrium strategy, which involves a system of FBSDEs. Second, by solving these FBSDEs, we obtain an equilibrium strategy in a linear feedback form of the surplus and the liability. Finally, we consider a Markovian case where the interest rate is given by the Vasiček model.