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Fully coupled forward-backward stochastic differential equations and applications to optimal control. (English) Zbl 0931.60048

Let (Ω,,P) be a probability space, and let {B t } t0 be a d-dimensional Brownian motion in this space; t 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):

x t =a+ 0 t b(s,x s ,y s ,z s )ds+ 0 t σ(s,x s ,y s ,z s )dB s ,
y t =Φ(x T )+ t T f(s,x s ,y s ,z s )ds- t T z s dB s ,t[0,T],

where (x,y,z) takes values in n × m × m+d , and b, f, σ are mappings with appropriate dimensions which are, for each fixed (x,y,z), t -progressively measurable. They are also Lipschitz with respect to (x,y,z); T>0 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.

60H10Stochastic ordinary differential equations
93E03General theory of stochastic systems