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 .

scholarly journals First exit time of a Lévy flight from a bounded region in ℝN

2015 ◽  
Vol 52 (3) ◽  
pp. 649-664 ◽  
Author(s):  
Yoora Kim ◽  
Irem Koprulu ◽  
Ness B. Shroff
Keyword(s):  
Exit Time ◽  
Length Distribution ◽  
Step Length ◽  
Upper And Lower Bounds ◽  
Bounded Region ◽  
Lévy Flight ◽  
First Exit Time ◽  
Levy Flight ◽  
Distribution Parameters ◽  
The Mean

In this paper we characterize the mean and the distribution of the first exit time of a Lévy flight from a bounded region inN-dimensional spaces. We characterize tight upper and lower bounds on the tail distribution of the first exit time, and provide the exact asymptotics of the mean first exit time for a given range of step-length distribution parameters.

scholarly journals AN INTERMEDIATE REGIME FOR EXIT PHENOMENA DRIVEN BY NON-GAUSSIAN LÉVY NOISES

2008 ◽  
Vol 08 (03) ◽  
pp. 583-591 ◽  
Author(s):  
ZHIHUI YANG ◽  
JINQIAO DUAN
Keyword(s):  
Dynamical System ◽  
Noise Intensity ◽  
Exit Time ◽  
First Exit Time ◽  
Small Noise ◽  
Intermediate Regime ◽  
Mean Exit Time ◽  
Small Intensity ◽  
The Mean ◽  
Non Gaussian

A dynamical system driven by non-Gaussian Lévy noises of small intensity is considered. The first exit time of solution orbits from a bounded neighborhood of an attracting equilibrium state is estimated. For a class of non-Gaussian Lévy noises, it is shown that the mean exit time is asymptotically faster than exponential (the well-known Gaussian Brownian noise case) but slower than polynomial (the stable Lévy noise case), in terms of the reciprocal of the small noise intensity.

scholarly journals First exit time of a Lévy flight from a bounded region in ℝN

2015 ◽  
Vol 52 (03) ◽  
pp. 649-664 ◽  
Author(s):  
Yoora Kim ◽  
Irem Koprulu ◽  
Ness B. Shroff
Keyword(s):  
Exit Time ◽  
Length Distribution ◽  
Step Length ◽  
Upper And Lower Bounds ◽  
Bounded Region ◽  
Lévy Flight ◽  
First Exit Time ◽  
Levy Flight ◽  
Distribution Parameters ◽  
The Mean

In this paper we characterize the mean and the distribution of the first exit time of a Lévy flight from a bounded region in N-dimensional spaces. We characterize tight upper and lower bounds on the tail distribution of the first exit time, and provide the exact asymptotics of the mean first exit time for a given range of step-length distribution parameters.

scholarly journals On first exit times and their means for Brownian bridges

2019 ◽  
Vol 56 (3) ◽  
pp. 701-722 ◽  
Author(s):  
Christel Geiss ◽  
Antti Luoto ◽  
Paavo Salminen
Keyword(s):  
Brownian Bridge ◽  
Exit Time ◽  
Three Dimensional ◽  
Exit Times ◽  
First Exit Time ◽  
Binomial Tree ◽  
Brownian Bridges ◽  
Bessel Bridge ◽  
The Mean ◽  
First Exit Times

AbstractFor a Brownian bridge from 0 to y, we prove that the mean of the first exit time from the interval $\left( -h,h \right),h>0$ , behaves as ${\mathrm{O}}(h^2)$ when $h \downarrow 0$ . Similar behaviour is also seen to hold for the three-dimensional Bessel bridge. For the Brownian bridge and three-dimensional Bessel bridge, this mean of the first exit time has a puzzling representation in terms of the Kolmogorov distribution. The result regarding the Brownian bridge is applied to provide a detailed proof of an estimate needed by Walsh to determine the convergence of the binomial tree scheme for European options.

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