Let be a probability space, and let be a -dimensional Brownian motion in this space; denotes the natural filtration of this Brownian motion. The authors consider existence and uniqueness problems for the following fully coupled forward-backward stochastic differential equation (FBSDE):
where takes values in , and , , are mappings with appropriate dimensions which are, for each fixed , -progressively measurable. They are also Lipschitz with respect to ; is an arbitrarily prescribed number and the time interval is called the time duration. Finally, several examples of FBSDE related to stochastic optimal control and differential games problems are given.