using relevant linear forward-backward stochastic differential equations, it obtains a calculating formula of the retained proportion or retention for the reinsurance.

Starting from systematic view,the paper integrates compensations that insuers will be up against with its return on investment and establishes linear forward-backward stochastic differential equations for proportional and excess-of-loss reinsurance premiums.

The well-posedness of time-delayed forward-backward stochastic differential equations is studied.

In this paper, we investigate the nature of the adapted solution to a class of forward-backward stochastic differential equations (short for FBSDE) without the non-degenerate condition for the forward equation.

The existence and uniqueness for the solution of forward-backward doubly stochastic differential equations were obtained under local Lipschitz condition,where the time duration could be arbitrarily given.

A general type of forward-backward doubly stochastic differential equations(FBDSDEs in short) was studied, which extends many important equations well studied before, including stochastic Hamiltonian systems.