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.

scholarly journals FIRST EXIT TIMES OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY MULTIPLICATIVE LÉVY NOISE WITH HEAVY TAILS

2011 ◽  
Vol 11 (02n03) ◽  
pp. 495-519 ◽  
Author(s):  
ILYA PAVLYUKEVICH
Keyword(s):  
Dynamical System ◽  
Differential Equations ◽  
Heavy Tails ◽  
Multiplicative Noise ◽  
Exit Time ◽  
Lévy Noise ◽  
Exit Times ◽  
First Exit Time ◽  
First Exit Times ◽  
Levy Noise

In this paper, we study first exit times from a bounded domain of a gradient dynamical system Ẏt = -∇U(Yt) perturbed by a small multiplicative Lévy noise with heavy tails. A special attention is paid to the way the multiplicative noise is introduced. In particular, we determine the asymptotics of the first exit time of solutions of Itô, Stratonovich and Marcus canonical SDEs.

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.

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 Exact simulation of first exit times for one-dimensional diffusion processes

2020 ◽  
Vol 54 (3) ◽  
pp. 811-844
Author(s):  
Samuel Herrmann ◽  
Cristina Zucca
Keyword(s):  
Diffusion Processes ◽  
Exit Time ◽  
Convergent Series ◽  
Mathematical Finance ◽  
Exit Times ◽  
Rejection Sampling ◽  
Girsanov Transformation ◽  
One Dimensional ◽  
Usual Procedure ◽  
First Exit Times

The simulation of exit times for diffusion processes is a challenging task since it concerns many applications in different fields like mathematical finance, neuroscience, reliability… The usual procedure is to use discretization schemes which unfortunately introduce some error in the target distribution. Our aim is to present a new algorithm which simulates exactly the exit time for one-dimensional diffusions. This acceptance-rejection algorithm requires to simulate exactly the exit time of the Brownian motion on one side and the Brownian position at a given time, constrained not to have exit before, on the other side. Crucial tools in this study are the Girsanov transformation, the convergent series method for the simulation of random variables and the classical rejection sampling. The efficiency of the method is described through theoretical results and numerical examples.

scholarly journals First-passage time of Markov processes to moving barriers

1984 ◽  
Vol 21 (4) ◽  
pp. 695-709 ◽  
Author(s):  
Henry C. Tuckwell ◽  
Frederic Y. M. Wan
Keyword(s):  
Differential Equation ◽  
Markov Processes ◽  
First Passage Time ◽  
Exit Time ◽  
Passage Time ◽  
First Exit Time ◽  
First Passage ◽  
Sample Paths ◽  
First Exit Times ◽  
First Passage Time Problems

The first-passage time of a Markov process to a moving barrier is considered as a first-exit time for a vector whose components include the process and the barrier. Thus when the barrier is itself a solution of a differential equation, the theory of first-exit times for multidimensional processes may be used to obtain differential equations for the moments and density of the first-passage time of the process to the barrier. The procedure is first illustrated for first-passage-time problems where the solutions are known. The mean first-passage time of an Ornstein–Uhlenbeck process to an exponentially decaying barrier is then found by numerical solution of a partial differential equation. Extensions of the method to problems involving Markov processes with discontinuous sample paths and to cases where the process is confined between two moving barriers are also discussed.

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

1989 ◽  
Vol 21 (01) ◽  
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

LetTbe the first exit time of Brownian motionW(t) from a region ℛ ind-dimensional Euclidean space having a smooth boundary. Given points ξ0and ξ1in ℛ, ordinary and large-deviation approximations are given for Pr{T &lt; ε|W(0) = ξ0,W(ε)=ξ1} asε→ 0. Applications are given to hearing the shape of a drum and approximating the second virial coefficient.

scholarly journals Diffusive escape through a narrow opening: new insights into a classic problem

2017 ◽  
Vol 19 (4) ◽  
pp. 2723-2739 ◽  
Author(s):  
Denis S. Grebenkov ◽  
Gleb Oshanin
Keyword(s):  
Exit Time ◽  
Angular Size ◽  
First Exit Time ◽  
Classic Problem ◽  
The Mean ◽  
Narrow Opening

We study the mean first exit time (Tε) of a particle diffusing in a circular or a spherical micro-domain with an impenetrable confining boundary containing a small escape window (EW) of an angular size ε.

Sign in / Sign up

Export Citation Format

Share Document