Wiener Processes

The Wiener process is one of the easiest models for random accumulation of degradation over time. It is based on the assumption of an additive accumulation of degradation with linear wear intensity. Since the Wiener process is based on normally distributed increments, this chapter presents some basic properties of the Gaussian (normal) distribution. It provides basic properties of the well-known Brownian motion, which is a specific Wiener process. The chapter also presents three constructions of the Brownian motion, namely random walk approximation, Brownian bridge construction and Karhunen-Loeve theorem, leading to different simulation methods. A Wiener process is a Brownian motion with an additional linear drift function and an additional variance parameter. The chapter discusses the statistical inference for both degradation and time to failure data. The distribution of the time to failure can easily be expressed with respect to the inverse Gaussian distribution.