If $ M$ is a continuous local martingale, then it exists a sequence of partitions of $[0,\infty)$ with $|\Pi_n| \to 0 $ ($|\Pi|$ denotes the mesh side) such that
$ P(\lim_{n\to \infty} Q^{\Pi_n}_t = \langle M \rangle_t \mbox{ simulataneously for all t $\ge$ 0 }) = 1$
Where, $Q^{\Pi_n}_t:=\sum_{t_i \in \Pi_n} (M_{t_{i+1}\wedge t}-M_{t_i\wedge t})^2$ for $t\ge 0$.
I have two question about the proof:
- If we know, $\lim_{|\Pi|\to 0} Q_t^\Pi = \langle M \rangle_t$ in probability, we deduced that using a diagonal procedure we have a sequence $(\Pi_n)_n$ with
$P(\lim_{n\to \infty} Q^{\Pi_n}_q = \langle M \rangle_q \mbox{ simulataneously for all rational q $\ge$ 0 }) = 1$
Diagonal argument is clear. But here we use also the fact, that for a converging sequence in probability, there exists a subsequence which converges a.s., right?
- Why is this enough to show? They say, because $\langle M \rangle$ is continuous. Could someone explain this in more detail?
Thanks
math