Suppose we have two square integrable continuous-time martingales X,Y: $X=\{X_t,\mathcal{F}_t; 0\leq t <\infty\}$ and $Y=\{Y_t,\mathcal{F}_t; 0\leq t <\infty\}$.
If we consider the cases X=Y, or X is independent of Y, then we can easily see that XY is a submartingale (using Jensen's inequality in the former case and by splitting the conditional expectation as a product in the latter).
Since X=Y and X,Y independent are quite extreme cases, it leads me to suspect that in general the product of XY must be a submartingale. How could I go about to prove this (provided it is true)?
$(X_tY_t,t\geqslant 0)$ may not be a submartingale, for example when $Y_t=-X_t$ and $X_t$ is not constant.
We have the submartingale property (under the assumptions that $(X_t,t\geqslant 0)$ and $(Y_t,t\geqslant 0)$ are martingales) if and only if for each $t>s$, $$E\left[(X_t-X_s)(Y_t-Y_s))\mid\mathcal F_s\right]\geqslant 0.$$