Martingale representation for degenerate diffusions

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 $\sigma:[0,1]\times C([0,1],\R^n)\to \R^n\otimes \R^d$, $b:[0,1]\times C([0,1],\R^n)\to \R^n$, $B$ 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 $(\calF_t(X),t\in [0,1])$ can be represented as $$M_t=E[M_0]+\int_0^t P_s(X)\alpha_s(X).dB_s$$ where $\alpha(X)$ is an $\R^d$-valued process adapted to $(\calF_t(X),t\in [0,1])$, satisfying $E\int_0^t(a(X_s)\alpha_s(X),\alpha_s(X))ds<\infty$, $a=\sigma^\star\sigma$ and $P_s(X)$ denotes a measurable version of the orthogonal projection from $\R^d$ to $\sigma(X_s)^\star(\R^n)$. In particular, for any $h\in H$, we have \begin{equation} \label{wick} E[\rho(\delta h)|\calF_1(X)]=\exp\left(\int_0^1(P_s(X)\dot{h}_s,dB_s)-\half\int_0^1|P_s(X)\dot{h}_s|^2ds\right)\,, \end{equation} where $\rho(\delta h)=\exp(\int_0^1(\dot{h}_s,dB_s)-\half |H|_H^2)$. This result gives a new development as an infinite series of the $L^2$-functionals of the degenerate diffusions. We also give an adequate notion of "innovation process" associated to a degenerate diffusion which corresponds to the strong solution when the Brownian motion is replaced by an adapted perturbation of identity. This latter result gives the solution of the causal Monge-Amp\`ere equation.}