ABSTRACT
Then the family { st,0 s t < }, is termed a stochastic flow if it is obtained as a solution of a stochastic differential equation (SDE) and satisfying (i)-(iii):
d st = F( st,dt), ss(0,w) = x, (51)
where {F(x,t),t 0} is a family of random functions for each x, which is either a semi-martingale relative ito a filtration {Ft,t 0} from or more simply an L2,2-bounded process in the sense of Bochner, with F(x,0) = 0. This includes many types of flows, such as Brownian, martingale, or harmonizable classes. It is now necessary to give a meaning for the SDE (51), and as usual converting it into an integral equation written symbolically as:
st(x) = x + ts F( sr,dr), 0 s r < t, (52)
and this reduces to the usual Ito integral if F(Xt,dt) = dXt. The general case is obtained using linear approximaions to F under conditions which make this possible and (52) meaningful, as described below.
