6
$\begingroup$

In Stochastic Calculus for Finance II: Continuous-time Models by Steve Shreve,

Theorem 3.4.3. Let $W$ be a Brownian motion. Then $[W, W](T) = T$ for all $T > 0$ almost surely.

where $[W, W](T)$ is the quadratic variation of $W$ up to time T $ [W, W](T): = \lim_{||\Pi|| \to 0} \sum_{j=0}^{n-1} [W(t_{j+1}) - W(t_j)]^2 $ where $\Pi=[t_0, t_1, \dots, t_n] \text{ and } 0 \leq t_0 < t_1 < \cdots < t_n = T$ and $||\Pi||= \max_{j=0,\dots, n-1} (t_{j+1} -t_j).$

One fellow student also says a stronger conclusion is also true, i.e.

$[W, W](T)$ converges to $T$ in $L^2$ or in $L^p, p>1$, for all $T > 0$.

By "stronger", I mean I heard it implies the original theorem for convergence a.e..

I wonder if there are some texts or online tutorials for proving this stronger conclusion, and/or if you could kindly provide the proof here?

Thanks!

  • 0
    @George: Thank you so much! Yes, it does clarify my misunderstanding. If you could post a full answer, that will be great, and I can upvote it.2011-11-08

0 Answers 0