I kindly ask for your help in the matter of usage of articles. My question is the following. I usually see no article in front of "Brownian motion", e.g. (introducing a process)
B is Brownian motion
I assume this is the right way to say. Am I right? However, I saw multiple different articles in front of "fractional Brownian motion" and its shorthand "fBm". So is it correct to write:
- M is a fBm
- M is an fBm
- M is the fBm
- M is fBm
It is acceptable to use "$B$ is Brownian Motion", since it is typically understood that you are referring to the standard Brownian motion process. If you want to be more specific I have also seen "$B$ is the Brownian motion process" or "$B$ is the standard Brownian motion process".
Personally, my preference is to call the stochastic process $B(t)$ the Wiener Process instead of Brownian motion; Brownian motion is the physical process of particles in fluids that was observed by Robert Brown; $B(t)$ is the mathematical object constructed by Norbert Wiener (and others) to model Brownian motion. So I usually use the letter $W(t)$, and I would say "Let $W(t)$ be the Wiener Process".
With fractional Brownian motion, there is not a single standard fBm process (there is a parameter, $H$), so it would be more correct to say "Let $B_H(t)$ be a fBm with Hurst exponent $H$" or at first usage, "let $B_H(t)$ be a fractional Brownian motion (fBm) with Hurst exponent $H$".