Quadratic Variation and Brownian Motion

Question

Let $(X_n,F_n)$ be a martingale with $X_n \in L^2(\Omega,F,\mathbb{P})$. The quadratic Variation $(<X>_n)_n$ of the process $(X_n)_n$ is defined as $$ <X>_n := \sum\nolimits_{i=1}^{n}(\mathbb{E}[X_i^2|F_{i-1}]-X_{i-1}^2) $$

Show that $\mathbb{E}[<X>_n]=Var(X_n - X_0)$. Now let $X_n = W_{nh}$ for a Brownian Motion $(W_t)_t$ and $h>0$. Show that the mean of the quadratic Variation of $(X_n)_n$ is unbound for $n\to \infty$.

My Problem is more the second part, but if someone has an easy way for the Formular I would also very much appreciate it.

Answer 1

I think, I have an easy way to proof Formular I. Using the the law of total expectation:

$E <X>_n = \sum_{i=1}^n [E(E(X_i^2|F_{i-1}) - E(X_{i-1}^2)] = \sum_{i=1}^n [E(X_i^2) - E(X_{i-1}^2)] = E(X_n^2) - E(X_0^2)$

Finally, notice that:

$Var(X_n - X_0) = E(X_n - X_0)^2 = E(X_n^2) - 2E(X_n X_0) + E(X_0^2) = E(X_n^2) - 2E(E(X_n X_0 | F_0)) + E(X_0^2) = E(X_n^2) - 2E(X_0 E(X_n|F_0) + E(X_n^2) = E(X_n^2) - E(X_0^2)$

For Brownian Motion, using that $W_n - W_0 \sim N(0, n)$, we have:

$E(<W_{nh}>) = Var(W_{nh} - X_0) = nh \to \infty, n \to \infty$