  This paper is the first part of a series of papers on filtering for partially
observed jump diffusions satisfying a stochastic differential equation driven
by Wiener processes and Poisson martingale measures. The coefficients of the
equation only satisfy appropriate growth conditions. Some results in filtering
theory of diffusion processes are extended to jump diffusions and equations for
the time evolution of the conditional distribution and the unnormalised
conditional distribution of the unobserved process at time $t$, given the
observations until $t$, are presented.
