I am trying to solve the following thing. I have two random processes $(X_t)$ and $(Y_t)$, on the interval $[0,T]$. For what is worth, $(X_t)$ is non-negative.
I want to show that $$ \lim_{\Delta_t\rightarrow 0}\sum_{i=0}^{n-1}\mathbb{E}\left[X_{t_{i+1}}\left(\int_{t_i}^{t_{i+1}}Y_tdt\right)^2\right] = 0 $$ (sum corresponding to a partition $0=t_0<t_1<\dots<t_n=T$, with $\Delta t=\max \Delta t_i$).
I have tried several approaches but can't just get there. I can show it without the expectation, i.e., pointwise, but using that $\max_{|a-b|\leq \Delta_t}|Y_a-Y_b|$ goes to 0 as $\Delta_t$ goes to $0$, but the proof doesn't seem to extend to the case with expectation.
Any help is appreciated, thanks!