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

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