We consider the process $dY_{t}=u_{t}\,dt+dW_{t},$
where $u$ is a process not
necessarily adapted to $\mathcal{F}^{Y}$ (the filtration generated by the
process $Y)$ and $W$ is a Brownian motion. We obtain a general
representation for the likelihood ratio of the law of the $Y$ process
relative to Brownian measure. This representation involves only one basic
filter (expectation of $u$ conditional on observed process $Y$). This
generalizes the result of Kailath and Zakai [Ann. Math. Statist.
42
(1971) 130-140] where it is assumed that
the process $u$ is adapted to $\mathcal{F}^{Y}.$ In particular, we consider
the model in which $u$ is a functional of $Y$ and of a random element $X$
which is independent of the Brownian motion $W.$ For example, $X$ could be a
diffusion or a Markov chain. This result can be applied to the estimation of
an unknown multidimensional parameter $\theta$ appearing in the dynamics of
the process $u$ based on continuous observation of $Y$ on the time interval
$[0,T]$. For a specific hidden diffusion financial model in which $u$ is an
unobserved mean-reverting diffusion, we give an explicit form for the
likelihood function of $\theta.$ For this model we also develop a
computationally explicit E--M algorithm for the estimation of $\theta.$ In
contrast to the likelihood ratio, the algorithm involves evaluation of a
number of filtered integrals in addition to the basic filter.