There are lots of questions on this site asking when Stochastic Integrals are Martingales. I have a similar question : Fix $0<T<\infty$. Given a Brownian Motion $W_t \in \mathbb{R}^d$ on a probability space $(\Omega,\mathcal{F}_t,\mathbb{P})$, let $\tau$ be a stopping time with respect to the filtration generated by $\{W_t , t\in [0,T]\}$.
For what $\sigma \in \mathbb{R}^{d \times d}$ is
$$ \int_0^{t\wedge \tau} \sigma_s dW_s $$
a Martingale ?
Question 2 : or a Local Martingale?
I am not so interested in Question 2, but it would be nice to know.
EDIT : From what I have read I know some sort of measurability assumptions on $\sigma$ is required, in addition I guess that something like $$ \mathbb{E}\Big[\int_0^{t\wedge \tau} \|\sigma_s \|^2 ds \Big]< \infty$$ might be enough? Could anyone please please clarify :). Where $\|\cdot\|$ is the usual Euclidean matrix norm.