Parameter estimation from observations of first-passage times of the Ornstein–Uhlenbeck process and the Feller process

Abstract

Renewal point processes show up in many different fields of science and engineering. In some cases the renewal points become the only observable parts of an anticipated hidden random variation of some physical quantity. The hypothesis might be that a hidden random process originating from zero or some other low value only becomes visible at the time of first crossing of some given value level, and that the process is restarted from scratch immediately after the level crossing. It might then be of interest to reveal the defining properties of this hidden process from a sample of observed first-passage times. In this paper the hidden process is first anticipated as a non-stationary Ornstein–Uhlenbeck (OU) process with unknown parameters that have to be estimated only by use of the information contained in a sample of first-passage times. The estimation method is a direct application of the Fortet integral equation of the OU process. A non-stationary Feller process is considered subsequently. As the OU process, the Feller process has a known transition probability distribution that allows the formulation of the integral equation. The described integral equation estimation method also provides a subjective graphical test of the applicability of the OU process or the Feller process when applied to a reasonably large sample of observed first-passage data. These non-stationary processes have several applications in biomedical research, for example as idealized models of the neuron membrane potential. When the potential reaches a certain threshold the neuron fires, whereupon the potential drops to a fixed initial value, from where it continuously builds up again until the next firing. Also in civil engineering there are hidden random phenomena such as internal cracking or corrosion that after some random time break through to the material surface and become observable. However, the OU process has as a model of physical phenomena the defect of not being bounded to the negative side. This defect is not present for the Feller process, which therefore may provide a useful modeling alternative to the OU process.