Problèmes de premier passage résolubles par la méthode de séparation des variables

In this paper, bidimensional stochastic processes defined by ax(t) = y(t)dt and dy(t) = m(y)dt + [2v(y)]1/2 dW(t), where W(t) is a standard Brownian motion, are considered. In the first section, results are obtained that allow us to characterize the moment-generating function of first-passage times for processes of this type. In Sections 2 and 5, functions are computed, first by fixing the values of the infinitesimal parameters m(y) and v(y) then by the boundary of the stopping region.

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Problèmes de premier passage résolubles par la méthode de séparation des variables

Journal of Applied Probability ◽

10.2307/3215292 ◽

1995 ◽

Vol 32 (2) ◽

pp. 337-348 ◽

Cited By ~ 1

Author(s):

Mario Lefebvre

Keyword(s):

Brownian Motion ◽

Stochastic Processes ◽

Generating Function ◽

Moment Generating Function ◽

First Passage Times ◽

Standard Brownian Motion ◽

First Passage ◽

The Moment ◽

Passage Times

In this paper, bidimensional stochastic processes defined by ax(t) = y(t)dt and dy(t) = m(y)dt + [2v(y)]1/2dW(t), where W(t) is a standard Brownian motion, are considered. In the first section, results are obtained that allow us to characterize the moment-generating function of first-passage times for processes of this type. In Sections 2 and 5, functions are computed, first by fixing the values of the infinitesimal parameters m(y) and v(y) then by the boundary of the stopping region.

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The logarithmic distribution of first passage times of standard Brownian motion

Trabajos de Estadistica y de Investigacion Operativa ◽

10.1007/bf03004953 ◽

1972 ◽

Vol 23 (1-2) ◽

pp. 125-128

Author(s):

L. K. Roy

Keyword(s):

Brownian Motion ◽

First Passage Times ◽

Standard Brownian Motion ◽

First Passage ◽

Logarithmic Distribution ◽

Passage Times

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On the Test and Power of Zero Drift on First Passage Times in Brownian Motion

Technometrics ◽

10.1080/00401706.1976.10489454 ◽

1976 ◽

Vol 18 (3) ◽

pp. 341-342 ◽

Cited By ~ 3

Author(s):

S. A. Patil ◽

J. L. Kovner

Keyword(s):

Brownian Motion ◽

First Passage Times ◽

Zero Drift ◽

First Passage ◽

Passage Times

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First passage times for a tracer particle in single file diffusion and fractional Brownian motion

The Journal of Chemical Physics ◽

10.1063/1.4707349 ◽

2012 ◽

Vol 136 (17) ◽

pp. 175103 ◽

Cited By ~ 26

Author(s):

Lloyd P. Sanders ◽

Tobias Ambjörnsson

Keyword(s):

Brownian Motion ◽

Fractional Brownian Motion ◽

Tracer Particle ◽

First Passage Times ◽

First Passage ◽

Single File ◽

Passage Times ◽

Single File Diffusion

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First-passage times of two-dimensional Brownian motion

Advances in Applied Probability ◽

10.1017/apr.2016.64 ◽

2016 ◽

Vol 48 (4) ◽

pp. 1045-1060 ◽

Cited By ~ 7

Author(s):

Steven Kou ◽

Haowen Zhong

Keyword(s):

Brownian Motion ◽

Analytical Solutions ◽

Laplace Transforms ◽

First Passage Times ◽

Absolute Difference ◽

Two Dimensional ◽

First Passage ◽

Quantitative Finance ◽

The 1960S ◽

Passage Times

AbstractFirst-passage times (FPTs) of two-dimensional Brownian motion have many applications in quantitative finance. However, despite various attempts since the 1960s, there are few analytical solutions available. By solving a nonhomogeneous modified Helmholtz equation in an infinite wedge, we find analytical solutions for the Laplace transforms of FPTs; these Laplace transforms can be inverted numerically. The FPT problems lead to a class of bivariate exponential distributions which are absolute continuous but do not have the memoryless property. We also prove that the density of the absolute difference of FPTs tends to ∞ if and only if the correlation between the two Brownian motions is positive.

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Best Tests for Zero Drift Based on First Passage Times in Brownian Motion

Technometrics ◽

10.1080/00401706.1973.10489016 ◽

1973 ◽

Vol 15 (1) ◽

pp. 125-132 ◽

Cited By ~ 16

Author(s):

Arthur Nádas

Keyword(s):

Brownian Motion ◽

First Passage Times ◽

Zero Drift ◽

First Passage ◽

Passage Times

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First passage times for Slepian process with linear and piecewise linear barriers

Extremes ◽

10.1007/s10687-021-00406-6 ◽

2021 ◽

Author(s):

Anatoly Zhigljavsky ◽

Jack Noonan

Keyword(s):

Brownian Motion ◽

Wiener Process ◽

Change Point ◽

Piecewise Linear ◽

Change Point Detection ◽

First Passage Times ◽

First Passage ◽

Explicit Formulas ◽

Point Detection ◽

Passage Times

AbstractIn this paper, we derive explicit formulas for the first-passage probabilities of the process S(t) = W(t) − W(t + 1), where W(t) is the Brownian motion, for linear and piece-wise linear barriers on arbitrary intervals [0,T]. Previously, explicit formulas for the first-passage probabilities of this process were known only for the cases of a constant barrier or T ≤ 1. The first-passage probabilities results are used to derive explicit formulas for the power of a familiar test for change-point detection in the Wiener process.

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Mean first-passage times of Brownian motion and related problems

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1952.0051 ◽

1952 ◽

Vol 211 (1106) ◽

pp. 431-443 ◽

Cited By ~ 42

Keyword(s):

Brownian Motion ◽

First Passage Time ◽

Passage Time ◽

First Passage Times ◽

Analytical Relationship ◽

First Passage ◽

Special Boundaries ◽

Mean First Passage Times ◽

Special Cases ◽

Passage Times

The theory of first-passage times of Brownian motion is developed in general, and it is shown that for certain special boundaries—the only ones of any importance—mean first-passage times can be derived very simply, avoiding the usual method involving series. Moreover, these formulae have a close analytical relationship to the better-known type of formulae for average 'displacements’ in given intervals; there exist certain pairs of reciprocal relations. Some new formulae, of mathematical interest, for translational Brownian motion are given. The main application of the general theory, however, lies in the derivation of experimentally particularly useful formulae for rotational Brownian motion. Special cases when external forces are present, and mean reciprocal first-passage times are discussed briefly, and finally it is shown how finite times of observation modify the mean first-passage time formulae of free Brownian motion.

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