We will define stochastic differential equations on manifolds in a way which will make them look as simi.lar to ordinary differential equations as possible. It may be helpful to keep in mind the fact that ordinary differential equations will be a special case of our stochastic differential equations.

The notation in this chapter i.s the same as before, but now we assume that we have a filtration {Ft:a < t < b} with b allowed to be infinite. In particular the σ-algebras Ft each contain all sets of measure zero in F. Throughout G and E will denote separable Banach spaces and H a separable Hilbert space, while

z:[a, b] → £o(Ω;E)

is sample continuous and satisfies condition A(q), q > 4, on each compact subset of [a, b). Some of the terminology from manifold theory and differential geometry is explained in Appendices A and B.

STOCHASTIC DYNAMICAL SYSTH1S

(A) Let M be a separable metrizable C3 manifold modelled on the Hilbert space H. Let Ḙ denote the trivial E-bundle, E x M → M, over M, and let TM be the tangent bundle of M. A stochastic dynamical system, (S.D.S.), (X, z) on M consists of a section X of ╙ (Ḙ, TM) for some E, together with a stochastic process z with values in E. In particular X(m) ∈ ╙ (E, TmM) for each m ∈ M. We will write X ∈ L (Ḙ, TM).

We shall only consider stochastic dynamical systems with X of class C1 and having locally Lipschitz first derivatives (in coordinate charts) e.g. X of class C2, and where z satisfies our standing hypotheses.