is product of two martingales a martingale with common filtration?

Question

hello please answer my question. 1)is product of two martingales respect of their natural filtration, martingale? 2)is product of two martingale respect to common filtration, martingale? thank for your helps

Answer 1

Almost never. An example: let $\{W_n\}$ be i.i.d with mean 0 , finite variance and $X_n=Y_n=W_1+W_2+...+W_n$. If $\{X_nY_n\}$ is a martingale the $E(X_nY_n)$ is independent of $n$. But $E(X_nY_n)=nEW_1^{2}$.