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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.


MSC:
60H10Stochastic ordinary differential equations
93E03General theory of stochastic systems