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 ${τ_n}$. The observed time series ${A(τ_n)}$ presents crude, delayed data. The crossing is first observed upon one of the observations, denoted $τ_ν$. 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, $[0, τ_{ν-1})$, or between the observations just before and just after the crossing, $[τ_{ν-1}, τ_ν)$, as a joint Laplace-Stieltjes transform and probability generating function of $A(τ_{ν-1})$, $A(τ_ν)$, $τ_{ν-1}$, and $τ_ν$. 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.