This is probably an easy question:
A local martingale is an adapted, cadlag process for which there is an increasing sequence of stopping times (going to $\infty$) such that the stopped process is a uniformly integrable martingale for each $n$.
This implies that each component is a local martingale.
Question: Is the converse also true? I.e. you have $M^1$ and $M^2$ as local martingales, is then also $(M^1,M^2)^\top$ a local martingale?
I don't think this is true. How would you construct such a sequence of stopping times that satisfies the definition above for all components? I'll think about a counterexample, but maybe someone knows better.