# Time Sensitive Analysis of Independent and Stationary Increment Processes

## TL;DR

This paper develops a new method to estimate the real-time behavior of jump processes with stationary increments from delayed observations, enabling better modeling of network crashes under cyber attacks.

## Contribution

It introduces a novel technique to reconstruct the paths of jump processes from delayed, discrete observations using joint Laplace-Stieltjes transforms and probability generating functions.

## Key findings

- Derived explicit joint distributions for process states and observation times.
- Applied the method to model and predict network crashes during cyber attacks.
- Enhanced understanding of process behavior from incomplete, delayed data.

## 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 ${\tau_n}$. 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.