Computation of time probability distributions for the occurrence of uncertain future events

The determination of the time at which an event may take place in the future is a well-studied problem in a number of science and engineering disciplines. Indeed, for more than fifty years, researchers have tried to establish adequate methods to characterize the behaviour of dynamic systems in time and implement predictive decision-making policies. Most of these efforts intend to model the evolution in time of nonlinear dynamic systems in terms of stochastic processes; while defining the occurrence of events in terms of first-passage time problems with thresholds that could be either deterministic or probabilistic in nature. The random variable associated with the occurrence of such events has been determined in closed-form for a variety of specific continuous-time diffusion models, being most of the available literature motivated by physical phenomena. Unfortunately, literature is quite limited in terms of rigorous studies related to discrete-time stochastic processes, despite the tremendous amount of digital information that is currently being collected worldwide. In this regard, this article provides a mathematically rigorous formalization for the problem of computing the probability of occurrence of uncertain future events in both discrete- and continuous-time stochastic processes, by extending the notion of thresholds in first-passage time problems to a fully probabilistic notion of “uncertain events” and “uncertain hazard zones”. We focus on discrete-time applications by showing how to compute those probability measures and validate the proposed framework by comparing to the results obtained with Monte Carlo simulations; all motivated by the problem of fatigue crack growth prognosis