Introduction to Stochastic Differential Equations with Applications to Modelling in Biology and Finance

11Introduction to the study of unidimensional Itô diffusions

11.1 The Ornstein–Uhlenbeck process and the Vasicek model

In Chapter  we saw that the projection into a coordinate axis of the Brownian motion of a particle suspended on a fluid was modelled by Einstein as a Wiener process of the form , where is the initial position of the particle and is the diffusion coefficient that gives the speed of change in the variance of the particle's position. This model, however, does not take into account that the friction is finite and so the particle, after a collision with a molecule of the fluid, does not change instantly its position and stop, but rather moves continuously with decreasing speed. In Einstein's model, the particle does not even have a defined speed (since the Wiener process does not have a derivative in the ordinary sense).

As an improvement to Einstein's model, the Ornstein–Uhlenbeck model (see Uhlenbeck and Ornstein (1930)) appeared in 1930. So far as I know, it was the first SDE to appear in the literature. In this model, it is assumed that the particle is subjected to two forces, the friction force and the force due to the random collisions with the fluid molecules. ...