local martingale?

Question

Consider the ito-process: $X(t)= \sigma(t) dW(t)$ for $t \in [0,T]$, where $\sigma$ is a predictable process and $\int_{0}^T \sigma^2 dt <\infty$.

Consider $X^2(t)$:

Appyling Ito's formula to $f(x)=x^2$ yields:$df(X(t))= 2 X(t) \sigma(t) dW(t) + \sigma^2(t) dt$

How can I see, that $2 X(t) \sigma(t) dW(t) $ is a local martingale?

Answer 1

$2 X_t \sigma(t) \; dW_t$ should be a local martingale because it is a stochatic integral with a martingale as it's integrator.