Probability densities for conditional statistics in the cubic sensor problem (Q1111240)

Consider the cubic sensor filtering problem \[ dX(t)=dW(t),\quad X(0)=X_ 0;\quad dY(t)=X^ 3(t)dt+dB(t),\quad Y(0)=0, \] where W and B are independent scalar Brownian motions and X(0) is a random variable independent of W and B with arbitrary distribution \(\pi_ 0.\) The author uses Malliavin calculus in an analogous way to his paper, J. Funct. Anal. 79, No.2, 288-331 (1988), to prove the following theorem: Let p(x,t/Y) be the unnormalized conditional density for the cubic sensor problem. For any \(n\in {\mathbb{N}}\) and linear independent set \[ \{\phi_ 1,\phi_ 2,...,\phi_ n\}\subset \{\phi | \phi (x)=\psi (x)e^{\beta | x|^ r},\quad \psi \in L^ 2({\mathbb{R}}^ n),\quad \beta \geq 0,\quad 0\leq r<4\} \] \[ let\quad \phi_ t^{(n)}(Y)=({\hat \phi}_ 1,...,{\hat \phi}_ n)\quad with\quad {\hat \phi}_ i=\int_{{\mathbb{R}}^ n}\phi_ i(x)\quad p(x,t,Y)dx. \] Then \(P_ 0(\phi_ t^{(n)})^{-1}\) is absolutely continuous with respect to the Lebesgue measure on \({\mathbb{R}}^ n\) for all t. This is also related to finite-dimensional computability of filters since it implies that, at any fixed time, the unnormalized conditional density cannot be characterized by a finite set of sufficient statistics.