  Ergodic properties of the signal-filtering pair are studied for continuous
time finite Markov chains, observed in white noise. The obtained law of large
numbers is applied to the stability problem of the nonlinear filter with
respect to initial conditions. The Furstenberg-Khasminskii formula is derived
for the top Lyapunov exponent of the Zakai equation and is used to estimate the
stability index of the filter.
