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.