What makes Itô processes special?

Question

As I understand it, Itô processes are those semi-martingales whose finite variation part is an integral (against Lebesgue measure) and whose (continuous local) martingale part is a stochastic integral against a Brownian motion. But I thought all (continuous local) martingales could be written as integrals against a Brownian motion, which would mean the only thing that distinguishes an Itô process among the semi-martingales is that its finite variation part is absolutely continuous. What am I missing?

Answer 1

To write a continuous local martingale as integral of a BM its filtration needs to be that of the BM we want to represent it with. That is an enormous restriction. See George Lowther's blog. The reason why Ito processes came to be is historical. At the time they were born semi martingales were not yet known. K. Ito choose the most natural definition of the time to build his theory on. Note that the condition on the filtration is more important than the continuity of the martingale. The latter follows automatically from that representation theorem.

For the sake of completeness I also mention another representation theorem which assumes that the martingale is continuous. Then, its filtration does not have to be Brownian. It can be enlarged to allow for a BM that represents the martingale as stochastic integral. See Karatzas & Shreve section 3.4.

Semimartingales were invented in the 1960ies by the Strasbourg school around P.A. Meyer and C. Dellacherie. That was a period of strong formalization not unlike what the Bourbaki group had done in other mathematical fields.