Maximum-likelihood estimation of a process with random transitions (failures)

A process with random transitions is represented by the difference equation
x_{n} = x_{n-1}+ u_{n}
where u
n
is a nonlinear function of a Gaussian sequence w_{n}. The nonlinear function has a threshold such that
u_{n} =0
for
|w_{n}| \leq W
. This results in a finite probability of no failure at every step. Maximum likelihood estimation of the sequence
X_{n}={x_{0},...,x_{n}}
given a sequence of observations
Y_{n} = { y_{1},...,y_{n} }
gives rise to a two-point boundary value (TPBV) problem, the solution of which is suggested by the analogy with a nonlinear electrical ladder network. Examples comparing the nonlinear filter that gives an approximate solution of the TPBV problem with a linear recursive filter are given, and show the advantages of the former. Directions for further investigation of the method are indicated.