Martingale representation for degenerate diffusions

Abstract

Let $(W,H,\mu )$ be the classical Wiener space on ${\mathbb{R}}^d$. Assume that $X=(X_t)$ is a diffusion process satisfying the stochastic differential equation $d{X}_t=\sigma (t,X)d{B}_t+b(t,X)dt$, where $\sigma :[0,1]\times C([0,1],{\mathbb{R}}^n)\to {\mathbb{R}}^n\otimes {\mathbb{R}}^d$, $b:[0,1]\times C([0,1],{\mathbb{R}}^n)\to {\mathbb{R}}^n$, $B$ is an ${\mathbb{R}}^d$-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 $$M_t=E[M_0]+{\int }_0^t{P}_s(X){\alpha }_s(X).d{B}_s$$ where $\alpha (X)$ is an ${\mathbb{R}}^d$-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 ${\mathbb{R}}^d$ to $\sigma (X_s{)}^{\star }({\mathbb{R}}^n)$. In particular, for any $h\in H$, we have \begin{equation} \label{wick} E[ρ(δh)|\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 $ρ(δh)=\exp(\int_0^1(\dot{h}_s,dB_s)-\half |H|_H^2)$. 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-Ampère equation.}

12 pages