4The Wiener process
4.1 Definition
The Wiener process plays an essential role in the stochastic differential equations we are going to study. It translates the cumulative effect (i.e. is the integral) of the underlying random perturbations affecting the dynamics of the phenomenon under study, so we are assuming that the perturbing noise is continuous‐time white noise. In practice, it is enough that this is a good approximation. As mentioned in Chapters and , Bachelier used the Wiener process in 1900 to model the price of stocks in the Paris stock market and Einstein used it to model the Brownian motion of a particle suspended on a fluid. However, only after 1920 was this process rigorously studied by Wiener and by Lévy. We will examine here the main properties. We will denote the standard Wiener process indifferently by 
(abbreviation of 
) or by 
(abbreviation of 
). Let us give the formal definition of the standard Wiener process.
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