Example of product of martingales which is not a martingale

Question

Let $(\Omega,F,\{F_n\},\mathbb P)$ a filtrated probability space and let $X$ and $Y$ $\{F_n\}$ martingales in $L^2$. I am looking for an example, that $XY$ is a sub martingale but not a martingale.(I know that one can easily show that this product has to be a sub martingale with Jensen's equality, i.e, take $X=Y$, but this attempt does not provide that the product is a martingale aswell..)

Any examples are welcome! (easy ones prefered)

Answer 1

Let $X_n$ be a simple random walk on the integers and take $Y_n = X_n$.

Answer 2

If $(X_n)_{n\ge 0}$ is a square-integrable martingale then $(X_n^2)_{n\ge 0}$ is a submartingale. It is a martingale if and only if $E((X_{n+1}-X_n)^2\mid\mathcal F_n)=0$ a.s for $n=0,1,2,\ldots$. This only happens in the trivial case $X_{n+1}=X_n$ a.s for all $n$.