scholarly journals On the excursions of drifted Brownian motion and the successive passage times of Brownian motion

2016 ◽  
Vol 457 ◽  
pp. 176-182 ◽  
Author(s):  
Mario Abundo
Keyword(s):  
Brownian Motion ◽  
Successive Passage ◽  
Passage Times

scholarly journals Local stationarity in exponential last-passage percolation

2021 ◽  
Author(s):  
Márton Balázs ◽  
Ofer Busani ◽  
Timo Seppäläinen
Keyword(s):  
Brownian Motion ◽  
Total Variation ◽  
High Probability ◽  
Probabilistic Methods ◽  
Side Length ◽  
Last Passage Percolation ◽  
Last Passage Times ◽  
Starting Point ◽  
Point To Point ◽  
Passage Times

AbstractWe consider point-to-point last-passage times to every vertex in a neighbourhood of size $$\delta N^{\nicefrac {2}{3}}$$ δ N 2 3 at distance N from the starting point. The increments of the last-passage times in this neighbourhood are shown to be jointly equal to their stationary versions with high probability that depends only on $$\delta $$ δ . Through this result we show that (1) the $$\text {Airy}_2$$ Airy 2 process is locally close to a Brownian motion in total variation; (2) the tree of point-to-point geodesics from every vertex in a box of side length $$\delta N^{\nicefrac {2}{3}}$$ δ N 2 3 going to a point at distance N agrees inside the box with the tree of semi-infinite geodesics going in the same direction; (3) two point-to-point geodesics started at distance $$N^{\nicefrac {2}{3}}$$ N 2 3 from each other, to a point at distance N, will not coalesce close to either endpoint on the scale N. Our main results rely on probabilistic methods only.

First-passage times of two-dimensional Brownian motion

2016 ◽  
Vol 48 (4) ◽  
pp. 1045-1060 ◽  
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.

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

1995 ◽  
Vol 32 (02) ◽  
pp. 337-348
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)]1/2 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.

Sufficient Conditions for a First Passage Time Process to be that of Brownian Motion

1969 ◽  
Vol 6 (01) ◽  
pp. 218-223
Author(s):  
M.T. Wasan
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
First Passage Time ◽  
Sufficient Conditions ◽  
Passage Time ◽  
Kolmogorov Equation ◽  
First Passage ◽  
State Variable ◽  
Strong Markov ◽  
Passage Times

In this paper we assign a set of conditions to a strong Markov process and arrive at a differential equation analogous to the Kolmogorov equation. However, in this case the duration variable is the net distance travelled and the state variable is a time, a situation precisely opposite to that of Brownian motion. Solving this differential equation under certain boundary conditions produces the density function of the first passage times of Brownian motion with positive drift (see [1]), with the aid of which we define a new stochastic process.

Density factorizations for brownian motion, meander and the three-dimensional bessel process, and applications

1984 ◽  
Vol 21 (3) ◽  
pp. 500-510 ◽  
Author(s):  
J.-P. Imhof
Keyword(s):  
Brownian Motion ◽  
Three Dimensional ◽  
Fixed Time ◽  
Bessel Process ◽  
Time Interval ◽  
Change Of Measure ◽  
Fixed Time Interval ◽  
Simple Change ◽  
Brownian Meander ◽  
Passage Times

Joint densities concerning in particular the value and time of the maximum over a fixed time interval, or the behavior over intervals determined by some first- and last-passage times, are determined for Brownian motion, the three-dimensional Bessel process and Brownian meander. Simple change of measure formulas permit easy passage from one process to the other. Examples are given.

The distribution of Brownian quantiles

1995 ◽  
Vol 32 (2) ◽  
pp. 405-416 ◽  
Author(s):  
Marc Yor
Keyword(s):  
Brownian Motion ◽  
Last Passage Times ◽  
Related Integral ◽  
Passage Times

The distribution of Brownian quantiles is determined, simplifying related integral expressions obtained by Lévy [9], [10] and more recently by Miura [11]. Three proofs are given, two of them involving last-passage times of Brownian motion, before time 1, at a given level.

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

Extremes ◽  
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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