Martingale representation for degenerate diffusions

摘要

Let (W, H, mu) be the classical Wiener space on R-d. Assume that X = (X-t) is a diffusion process satisfying the stochastic differential equation dX(t) = sigma(t, X)dB(t) + b(t, X)dt, where sigma : [0,1] x C([0, 1], R-n) -> R-n circle times R-d, b : [0, 1] x C([0,1], R-n) -> R-n, B is an R-d-valued Brownian motion. We suppose that the weak uniqueness of this equation holds for any initial condition. We prove that any square integrable martingale M w.r.t. to the filtration (F-t(X), t is an element of[0, 1]) can be represented as