  We study the behavior of independent and stationary increments jump processes
as they approach fixed thresholds. The exact crossing time is unavailable
because the real-time information about successive jumps is unknown. Instead,
the underlying process $A(t)$ is observed only upon a third-party independent
point process ${\tau_n}$. The observed time series ${A(\tau_n)}$ presents
crude, delayed data. The crossing is first observed upon one of the
observations, denoted $\tau_\nu$. We develop and further explore a new
technique to revive the real-time paths of $A(t)$ for all $t$ belonging to an
interval before the pre-crossing observation, $[0, \tau_{\nu-1})$, or between
the observations just before and just after the crossing, $[\tau_{\nu-1},
\tau_\nu)$, as a joint Laplace-Stieltjes transform and probability generating
function of $A(\tau_{\nu-1})$, $A(\tau_\nu)$, $\tau_{\nu-1}$, and $\tau_\nu$.
Joint probability distributions are obtained from the transforms in a tractable
form and they are applied to modeling of stochastic networks under cyber
attacks by accurately predicting their crash.
