Nonlinear filtering: The exact dynamical equations satisfied by the conditional mode

The signal x<inf>t</inf>is a stochastic process satisfying the stochastic differential equation<tex>dx = f(x)dt+dz</tex>. Observations<tex>\dot{y} =g(x) +\xi</tex>are taken, where<tex>\xi</tex>is white noise. The exact dynamical equation for the mode of the conditional density of x<inf>t</inf>is derived and discussed.