Abstract: The existence results of the optimal strategy are obtained for the stochastic recursive optimal control problem i.e. the cost function is described by the solution of a certain backward stochastic differential equation and the recursive mixed optimal control problem i.e. the controller has to decide the optimal stopping time also. The minimal and maximal mathematic expectations for the recursive optimal value function under one kind of equivalent probability measures set are also given in the paper.

Key words: Backward stochastic differential equation, Recursive optimal control, Mixed optimal control, Comparison theorem