Proving alternative notation of quadratic variation of Brownian motion

Question

Setting: Let $\{W_t,t\geq 0\}$ be a Brownian with respect to the standard filtration $\{\mathcal{F}_t,t\geq 0\}$.

Problem: We fix $t>0$ and must prove that:

$\left \langle W,W \right\rangle^{(n)}_{t}=\sum^n_{j=1}(W_{\frac{tj}{n}}-W_{\frac{t(j-1)}{n}})^2$

Can be written as:

$\left \langle W,W \right \rangle^{(n)}_{t}=\frac{t}{n}\sum^n_{i=1}X_i^2$

Where $X_i$ are independent identically distributed and have a standard normal distribution.

Question: I've attempted to solve this exercise, obviously, but kept getting stuck because I think my approach is wrong. I tried writing out the sum and using telescoping but I can't seem to get factors to drop out. Writing out the first sum does yield some similar factors but the square prohibits them from dropping off against eachother. Writing out the square and then applying telescoping also provides no luck. If I can't use telescoping for this problem I'm not sure how to approach it. Any tips or advice on solving this problem?

Any help is appreciated. Thanks!

Answer 1

Hints:

1. Show that $$\langle W,W \rangle_t^{(n)} = \sum_{j=1}^n Y_j^2$$ for independent Gaussian random variables $Y_j$, $j=1,\ldots,n$. (Hint to find $Y_j$: Do not expand the square! Look at the very definition of $\langle W,W \rangle_t^{(n)}$ ...)
2. Show that $\mathbb{E}(Y_j)=0$ and $\mathbb{E}(Y_j^2) = t/n$ for all $j=1,\ldots,n$.
3. Define $$X_j := \sqrt{\frac{n}{t}} Y_j, \qquad j=1,\ldots,n.$$ Using Step 2 prove that the so-defined random variables are independent and Standard Gaussian.
4. Use Step 1 to show that $$\langle W,W\rangle_t^{(n)} = \frac{t}{n} \sum_{j=1}^n X_j^2.$$

Answer 2

Just define $X_i= {\sqrt {\frac n t}} (W_{\frac {tj} n}-W_{\frac {t(j-1)} n})$. Use the fact that $(W_t)$ has independent increments and $W_{t+s}-W_t$ has $N(0,s)$ distribution.