Hello,
Not clear, what is in your example ?
Also, it's not that <M>_t that is equal to M_t^2, it's their expectations ! How can you make such a mistake while you're already studying BM's ?
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
The second equality follows from the fact that [M] is the unique increasing process for which M^2-[M] is a (local) martingale, which in your case as M is square integrable is a true martingale.
There is no easy way of explicitly working out the quadratic variation, but if you can write it as an SDE in terms of a BM then your task is easy.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