Maximum-likelihood estimation of a process with random transitions

A process with random transitions is represented by the difference equation xn=xn-1+un where un is a nonlinear function of a gaussian sequence wn. The nonlinear function has a threshold such that un=0 for |wn|?? W . This results in a finite probability of no failure at every step. Maximum likelihood estimation of the sequence Xn ={x0,...,xn} given a sequence of observations Yn= {y1,...,yn} 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.