Estimate for an expectation

Hello,

I have the following situation: Two stochastic processes X and Y. For X, I know that $\displaystyle E[|X_s]^2]\leq C$ for all $\displaystyle s\in[t,T]$. Now I need an estimate for the expression: $\displaystyle E[|X_sY_s|^2$, aim is to get rid of the X. Something like $\displaystyle E[|X_sY_s|^2]\leq E[|X_s|^2]E[|Y_s|^2]$ (does this hold???). Then I would obtain $\displaystyle E[|X_sY_s|^2]\leq C E[|Y_s|^2]$. Then I would be happy.

I hope, somebody has an idea.

Thanks in adavance,

twingeling