Q: is an martingale. Show that if then a.s.

Here is the answer:

a.s.

Couple of questions : 1. How do you justify the second inequality (I know how to show the first). how is this equal to ??

2. Is there an "easy" way of computing the quadratic variation of stochastic processes. For example the q.v. of a standard BM is t, and I understand the derivation, but it's using the definition! For that matter martingales, semi-marts or any other useful stochastic process......always using the definition? Thanks