Abstract
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 {τn}. The observed time series {A(τn)} presents crude, delayed data. The crossing is first observed upon one of the observations, denoted τν. 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,τν−1), or between the observations just before and just after the crossing, [τν−1,τν), as a joint Laplace–Stieltjes transform and probability generating function of A(t), A(τν−1), A(τν), τν−1, and τν. 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.
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