Hi, I've been working through various derivations of the stochastic calculus (mainly Rogers & Williams) and have been stuck on one point for a while now.
If:
M^2 is the Hilbert Space of square-integrable martingales. (Null at zero)
cM^2 is the subspace with continuous sample paths (a.s).
dM^2 is the orthogonal complement of cM^2 in M^2.
UI M is the space of uniformly integrable martingales.
Then the subspace of IV M^2 of square-integrable martingales of integrable variation is dense in dM^2.
Rogers and Williams use the following Theorem to justify this, but I just can't make the link!
Let M belong to M^2. Then M belongs to cM^2 iff MN belongs to UI M for every N in IV M^2.
If anyone would be able to help me on this I'd be really grateful..
Thanks.