  This paper is concerned with nonlinear filtering of the coefficients in asset
price models with stochastic volatility. More specifically, we assume that the
asset price process $ S=(S_{t})_{t\geq0} $ is given by \[
dS_{t}=r(\theta_{t})S_{t}dt+v(\theta_{t})S_{t}dB_{t}, \] where
$B=(B_{t})_{t\geq0}$ is a Brownian motion, $v$ is a positive function, and
$\theta=(\theta_{t})_{t\geq0}$ is a c\'{a}dl\'{a}g strong Markov process. The
random process $\theta$ is unobservable. We assume also that the asset price
$S_{t}$ is observed only at random times $0<\tau_{1}<\tau_{2}<....$ This is an
appropriate assumption when modelling high frequency financial data (e.g.,
tick-by-tick stock prices).
  In the above setting the problem of estimation of $\theta$ can be approached
as a special nonlinear filtering problem with measurements generated by a
multivariate point process $(\tau_{k},\log S_{\tau_{k}})$. While quite natural,
this problem does not fit into the standard diffusion or simple point process
filtering frameworks and requires more technical tools. We derive a closed form
optimal recursive Bayesian filter for $\theta_{t}$, based on the observations
of $(\tau_{k},\log S_{\tau_{k}})_{k\geq1}$. It turns out that the filter is
given by a recursive system that involves only deterministic Kolmogorov-type
equations, which should make the numerical implementation relatively easy.
