Life expectancy: Lower and upper bounds from surviving fractions and remaining life expectancy

The authors discuss one type of general forward-backward stochastic differential equations (FBSDEs) with It's stochastic delayed equations as the forward equations and anticipated backward stochastic differential equations as the backward equations. The existence and uniqueness results of the general FBSDEs are obtained. In the framework of the general FBSDEs in this paper, the explicit form of the optimal control for linearquadratic stochastic optimal control problem with delay and the Nash equilibrium point for nonzero sum differential games problem with delay are obtained.