Hallo,

I have to show that the following sets of stochastic processes are not equal:

$\displaystyle L^2:=\{(X_t)|(X_t)$ is progessive measurable and $\displaystyle \mathbb{E}[\int\limits_0^T X_t^2dt]< \infty \}$

$\displaystyle H^2:=\{(X_t)|(X_t)$ is progessive measurable and $\displaystyle \int\limits_0^T X_t^2dt< \infty$ a.s.$\displaystyle \}$

I know that these sets aren't equal.

But how can I show that $\displaystyle \int\limits_0^T X_t^2dt< \infty$ a.s. doesn't imply $\displaystyle \mathbb{E}[\int\limits_0^T X_t^2dt]< \infty $

Does anyone know a proof?

I also know that the Itô-integral $\displaystyle \int\limits_0^T X_t dW_t$ where $\displaystyle W_t$ is a standard Brownian motion and $\displaystyle X_t \in L^2$ is not just a local martingale, it's also a martingale.

But is $\displaystyle \int\limits_0^T X_t dW_t$ also a martingale if $\displaystyle X_t \in H^2$?

Can anybody help me?

Thanks!