Necessary conditions in stochastic linear quadratic problems and their applications

Stochastic linear quadratic optimal control problems are considered. A unified approach is proposed to treat the necessary optimality conditions of closed-loop optimal strategies and open-loop optimal controls. Notice that the former notion does not rely on initial wealth, while the later one does. Our conclusions of closed-loop optimal strategies are directly derived by suitable variational methods, the approach to which is different from [12], [11]. Moreover, the necessary conditions for closed-loop optimal strategies happen to be sufficient which takes us by surprise. Finally, two applications are given as illustration.