If I have two RCLL martingales $X,Y$, both bounded in $L^2$, hence uniformly integrable. Then we call $X,Y$ weakly orthogonal if $E[X_\infty Y_\infty]=0$ and we call $X,Y$ strongly orthogonal if $XY$ is a martingale. Now I have some question about this:
- Why is the product $XY$ uniformly integrable?
- If I know that $X^\tau Y^\tau$ is a martingale for every stopping time $\tau$, then $XY$ is martingale. Is this just because I can take $\tau:=\infty$ which is a stopping time, since it is constant to obtain the result?
Thanks in advance for your help.