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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 mean first passage time from node i to node j is the expected number of steps it takes for the process to reach node j from node i for the first time: (,) = = (,,) where P(i,j,r) denotes the probability that it takes exactly r steps to reach j from i for the first time.
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]
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 ...
The mean first passage time satisfies =. This can be used to calculate, for example, the time it takes for a Brownian motion particle in a box to hit the boundary of the box, or the time it takes for a Brownian motion particle in a potential well to escape the well.
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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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.