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

A process with random transitions is represented by the difference equation where unis a nonlinear function of a Gaussian sequence w_{n}. The nonlinear function has a threshold such that for . This results in a finite probability of no failure at every step. Maximum likelihood estimation of the sequence given a sequence of observations 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.