Let $(X_t)_{t\geq0}$ by a jump-diffusion process and define $\tau = \inf\{t\geq 0 : X_t \geq B \}$ the first time that $X_t$ exceeds $B$. Let $C>0$. I am having trouble understanding intuitively what is the meaning of the expectation:
$$ E[X_\tau \mathbb{1}_{\tau \leq C}] $$
Where $X_t$ is stopped at $\tau$. I know I haven't given any details about the process $X_t$, but what would be the general guidelines for how to calculate such an expectation?