I was wondering how to evaluate the following expression: \begin{equation} \mathbb{E}[\int_t^T\Delta_s dX_s \; |\{X_u\}_{t \leq u \leq T}] \end{equation}
where the stochastic process $X$ is a martingale, and $\Delta$ is adapted.
I know that without the path of $X$, this would just be $0$, but now that its path is given, I am unsure how I can manipulate this expression. $T$ can be any time increment so $T = t + dt$ is also a possible choice, but I couldn't get anywhere conclusive using that. I am also wondering if there are certain conditions under which this expression would evaluate to $0$ (other than $\Delta = 0$), since for the problem I am working on that would be 'ideal'.
Thanks!