Is any FV-Process a special Semimartingale? Any FV-Process can be represented as the difference of two increasing (or decreasing) processes and so any FV-Process is a quasimartingale. Due to Raos Theorem any FV-Process is a special semimartingale.
Is this statement correct or did I make a mistake? If it's correct, can this statement be deducted easier or is it just trivial and I haven't seen this?
I would be very grateful if you helped me.
Thanks in advance.
 A: You are right that as any adapted process $X$ of finite variation can be written as the difference of two adapted increasing processes, it holds that any such process $X$ is a quasimartingale. However, I don't know what theorem of Rao you refer to in order to conclude from this that $X$ is a special semimartingale.
A special semimartingale $X$ is a process which can be written as $X = X_0 + M + A$, where $M$ is a local martingale with initial value zero and $A$ is a predictable process of finite variation. It is thus immediate that any $\textbf{predictable}$ process $X$ of finite variation is a special semimartingale.
Alternatively, the following also holds. If $X$ is a process of finite variation which is also $\textbf{locally integrable}$, then $X$ is a speical semimartingale. This is because any process of finite variation is a semimartingale, and any semimartingale which is locally integrable is a special semimartingale. This latter claim can be found as Theorem 1 here:
http://almostsure.wordpress.com/2011/10/03/special-semimartingales/
By this theorem, a semimartingale $X$ is special if and only if it is locally integrable. Thus, an adapted process of finite variation which is not locally integrable cannot be a special semimartingale. As there exists adapted processes of finite variation which are not locally integrable, it does not in general hold that such processes are special semimartingales.
Thus, to sum up:
-Any predictable FV process is a special semimartingale.
-Any locally integrable FV process is a special semimartingale.
-It does not hold in general that adapted FV processes are special semimartingales.
