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Calculation of failure level based on inverse first passage problem

In real cases, the definition of failure sometimes seems ambiguous although boundary-crossing failures are widely accepted in data-analysis work. Usually we don't have really observed “real” failures, but we think a failure occurs once the system degradation index reaches a preset failure level. From this view, the failure level defines the failure when considering boundary-crossing failures. This point is emphasized in this paper and we want to reproduce the failure level based on inverse first passage problem (IFPT). Besides those technical results in this paper, an interesting conclusion is we actually have distinguished between “real” failures and boundary-crossing failures in data analysis, and this paper provided a way to model the “real” failures by first passage failures. The core issue discussed in this paper is called inverse first passage problem: supposing that the degradation dynamic is given whose failure is defined as first passage failure, and also the distribution of failure time is given or estimated from real data, this paper is to find the passage level under which the first passage density function is just the given distribution of failure time. Specifically, the degradation model considered in this paper is a time-dependent Ornstein-Uhlenbeck (OU) process which is verified in a real case before. Later based on a connection between the time-dependent OU process and Brownian motion, we here have derived a numerical algorithm to calculate the corresponding passage boundaries based on given first passage density. These results were tested for a fitted model where the first passage density function is initially calculated based on a given passage level, then we reproduced the same passage level based on our algorithm.