Is every continuous process an Ito process

Question

I see sometimes in Financial Maths literature that we have covered all continuous processes when assuming the following dynamics: $dX_t = \mu_t \, dt + \sigma_t \, dW_t$.

I can formulate my question in two ways and I am not sure they are entirely equivalent:

1. Is every continuous process an Ito process?
2. Can every continuous process be decomposed in a Brownian part and a finite variation part (Semimartingale)?

Thanks for your help.

Answer 1

In general, continuous processes fail to be semimartingales (and hence Itô processes). A classical example is the fractional Brownian motion with Hurst index $H \neq \frac{1}{2}$. The process is a Gaussian process and has continuous sample paths, but it fails to be a semimartingale (see this question).

Financial Mathematics tends to ignore the fact that the world is not governed by Brownian motion (and not even by the much larger class of continuous processes)...