On the Density of the First Exit Time of a Stable Process Across a Nonlinear Boundary

1986 ◽  
Vol 30 (2) ◽  
pp. 422-427
Author(s):  
F. G. Ragimov
Keyword(s):  
Nonlinear Boundary ◽  
Exit Time ◽  
Stable Process ◽  
First Exit Time

n-point, win-by-k games

1989 ◽  
Vol 26 (4) ◽  
pp. 807-814 ◽  
Author(s):  
Kyle Siegrist
Keyword(s):  
Exit Time ◽  
Bernoulli Trials ◽  
Expected Number ◽  
First Exit Time ◽  
World Series ◽  
Time Space ◽  
Gambler’S Ruin ◽  
Gambler's Ruin Problem ◽  
Positive Integers ◽  
Made In

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.

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

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 .

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

1989 ◽  
Vol 21 (1) ◽  
pp. 1-19 ◽  
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.

The first exit time; energy problems

2009 ◽  
pp. 268-286
Author(s):  
Pierre Cartier ◽  
Cecile DeWitt-Morette
Keyword(s):  
Exit Time ◽  
First Exit Time

The first exit time for a Bessel process from the minimum and maximum random domains

2009 ◽  
Vol 79 (20) ◽  
pp. 2115-2123 ◽  
Author(s):  
Lixin Song ◽  
Dawei Lu ◽  
Jinghai Feng
Keyword(s):  
Exit Time ◽  
Bessel Process ◽  
First Exit Time ◽  
Random Domains
Sign in / Sign up

Export Citation Format

Share Document