Time sensitive analysis of independent and stationary increment processes

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(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.

Current Trends in Random Walks on Random Lattices

“…This approach introduces some insurmountable uncertainty dependent on the crudeness of the observation process {τ n } ∞ n=1 . A sequence of papers, Dshalalow and his collaborators [50,68,69,[73][74][75][76]113] pursue methods referred to as time sensitive analysis that try to offer more precise look into the intermediate time period between the pre-exit observation and post-exit observation times during which the real time exit actually occurs, to glean some further insights about the process upon its exit.…”

Time Sensitive Analysis of Antagonistic Stochastic Processes and Applications to Finance and Queueing

“…In our next efforts to unfold Φ ν1<ν2 we will apply methods of discrete and continuous operational calculus. We will use an approach previously developed by Dshalalow [14] and further explored in Dshalalow and White [21] and referred to as a stochastic expansion. To this end, we define two sequences of exit indices…”

“…The main focus of our work is on time sensitive fluctuations, meaning that at least some of the component processes are with continuous time parameter. Some prior work on time sensitive fluctuations is due to Dshalalow [14] (pertaining to antagonistic games), Dshalalow and Nandyose [17,20] about one-dimensional processes, Dshalalow and White [21] on twodimensional processes. Dshalalow and White [21] studied random measures of two active nonnegative components competing against each other.…”

“…Some prior work on time sensitive fluctuations is due to Dshalalow [14] (pertaining to antagonistic games), Dshalalow and Nandyose [17,20] about one-dimensional processes, Dshalalow and White [21] on twodimensional processes. Dshalalow and White [21] studied random measures of two active nonnegative components competing against each other. However, the game was observed over arbitrary times (not adapted to the process) and consequently, the information on the status of the game was delayed.…”

“…However, the game was observed over arbitrary times (not adapted to the process) and consequently, the information on the status of the game was delayed. Nor did the authors of [21] consider a third, passive, non monotone component. The delayed factor yielded less tame functionals than in the present work.…”