The First Exit Time Stochastic Theory Applied to Estimate the Life-Time of a Complicated System

Consider a sequence of Bernoulli trials between players A and B in which player A wins each trial with probability p∈ [0, 1]. For positive integers n and k with k ≦ n, an (n, k) contest is one in which the first player to win at least n trials and to be ahead of his opponent by at least k trials wins the contest. The (n, 1) contest is the Banach match problem and the (n, n) contest is the gambler's ruin problem. Many real contests (such as the World Series in baseball and the tennis game) have an (n, 1) or an (n, 2) format. The (n, k) contest is formulated in terms of the first-exit time of the graph of a random walk from a certain region of the state-time space. Explicit results are obtained for the probability that player A wins an (n, k) contest and the expected number of trials in an (n, k) contest. Comparisons of (n, k) contests are made in terms of the probability that the stronger player wins and the expected number of trials.

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A variation on the Donsker–Varadhan inequality for the principal eigenvalue

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2016.0877 ◽

2017 ◽

Vol 473 (2204) ◽

pp. 20160877

Author(s):

Jianfeng Lu ◽

Stefan Steinerberger

Keyword(s):

Lower Bound ◽

Exit Time ◽

Principal Eigenvalue ◽

Short Paper ◽

First Eigenvalue ◽

Drift Diffusion ◽

First Exit Time ◽

The First Eigenvalue ◽

Order Elliptic Operator ◽

The Mean

The purpose of this short paper is to give a variation on the classical Donsker–Varadhan inequality, which bounds the first eigenvalue of a second-order elliptic operator on a bounded domain Ω by the largest mean first exit time of the associated drift–diffusion process via λ 1 ≥ 1 sup x ∈ Ω E x τ Ω c . Instead of looking at the mean of the first exit time, we study quantiles: let d p , ∂ Ω : Ω → R ≥ 0 be the smallest time t such that the likelihood of exiting within that time is p , then λ 1 ≥ log ( 1 / p ) sup x ∈ Ω d p , ∂ Ω ( x ) . Moreover, as p → 0 , this lower bound converges to λ 1 .

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Approximate exit probabilities for a Brownian bridge on a short time interval, and applications

Advances in Applied Probability ◽

10.2307/1427195 ◽

1989 ◽

Vol 21 (1) ◽

pp. 1-19 ◽

Cited By ~ 7

Author(s):

H. R. Lerche ◽

D. Siegmund

Keyword(s):

Virial Coefficient ◽

Large Deviation ◽

Brownian Bridge ◽

Exit Time ◽

Second Virial Coefficient ◽

Dimensional Euclidean Space ◽

Time Interval ◽

Short Time Interval ◽

First Exit Time ◽

Short Time

Let T be the first exit time of Brownian motion W(t) from a region ℛ in d-dimensional Euclidean space having a smooth boundary. Given points ξ0 and ξ1 in ℛ, ordinary and large-deviation approximations are given for Pr{T < ε |W(0) = ξ0, W(ε) = ξ 1} as ε → 0. Applications are given to hearing the shape of a drum and approximating the second virial coefficient.

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