Abstract: We consider a stochastic nonlinear dynamical process with annihilation of particles. This process can be viewed as the continuous time version of the extended
Kalman ﬁlter/smoother. It also plays an important role in stochastic optimal control theory. We derive a Gaussianapproximation for this process. With the use
of the path integral formalism we derive Euler-Lagrange equations for the mode.
Furthermore, we derive a linear noise approximation to estimate the size of the
ﬂuctuations around the mode, and estimates of the partition function, based on the
mode and Gaussian corrections. Numerical experiments conﬁrm the validity of
the approximation method. In addition, they show that the Gaussian correction
provides a signiﬁcant improvement of the estimate of the partition function.