Product of independent continuous local martingales is local martingale

Question

Revuz-Yor's book mentioned if $M$ and $N$ are independent continuous local martingales, then $MN$ is still local martingale. But I don't know how to prove it. Any help, thanks!

Answer 1

First of all, note that you should always specify the filtration since the answer depends, in general, on the given filtration. The following statements are both correct:

1. Let $(X_t,\mathcal{F}_t^X)_{t \geq 0}$, $(Y_t,\mathcal{F}_t^Y)_{t \geq 0}$ be continuous local martingales with respect to their natural filtration. Then $(X_t Y_t, \mathcal{F}_t^{X Y})_{t \geq 0}$ is a local martingale.
2. Let $(X_t,\mathcal{F}_t)_{t \geq 0}$, $(Y_t,\mathcal{F}_t)_{t \geq 0}$ be continuous local martingales. Then $(X_t Y_t, \mathcal{F}_t)_{t \geq 0}$ is a local martingale.

The continuity of the processes is crucial; the statements do not hold if the martingales are not continuous. For proofs see e.g. this (quite accessible) article by Cherny.