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[2] [3] [4] Modeling the probability of financial ruin as a first passage time was an early application in the field of insurance. [5] An interest in the mathematical properties of first-hitting-times and statistical models and methods for analysis of survival data appeared steadily between the middle and end of the 20th century.
The first exit time (from A) is defined to be the first hit time for S \ A, the complement of A in S. Confusingly, this is also often denoted by τ A. [1] The first return time is defined to be the first hit time for the singleton set {X 0 (ω)}, which is usually a given deterministic element of the state space, such as the origin of the ...
2.2 Lifetime distribution function and event density. ... the survival event density function is known as the first passage time density.
First passage percolation is one of the most classical areas of probability theory. It was first introduced by John Hammersley and Dominic Welsh in 1965 as a model of fluid flow in a porous media. [1] It is part of percolation theory, and classical Bernoulli percolation can be viewed as a subset of first passage percolation.
This is the smallest time after the initial time t 0 that y(t) is equal to one of the critical values forming the boundary of the interval, assuming y 0 is within the interval. Because y(t) proceeds randomly from its initial value to the boundary, τ(y 0) is itself a random variable. The mean of τ(y 0) is the residence time, [1] [2]
This distribution appears to have been first derived in 1900 by Louis Bachelier [6] [7] as the time a stock reaches a certain price for the first time. In 1915 it was used independently by Erwin Schrödinger [4] and Marian v. Smoluchowski [5] as the time to first passage of a Brownian motion.
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The mean sojourn time (or sometimes mean waiting time) for an object in a dynamical system is the amount of time an object is expected to spend in a system before leaving the system permanently. This concept is widely used in various fields, including physics, chemistry, and stochastic processes, to study the behavior of systems over time.