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I have read an article about semimartingales on Wikipedia and it says that: "A Brownian motion is a semimartingale". However, it is hard for me to find any proof of this statement, and I really doubt about its accuracy. Can any one verify if the statement is true, or show me some counter examples if the statement is wrong?
Here is the link of the article: https://en.wikipedia.org/wiki/Semimartingale
A semimartingale $X_t$ is a process that can be written as $$X_t = X_0 + M_t + A_t$$ where $M_t$ is a (local) martingale and $A_t$ is a process of finite variation. In particular, any martingale is a semimartingale, simply by setting $A_t = 0$. Since Brownian motion is a martingale, it is a semimartingale.
