Local Times and Excursion Theory for Brownian Motion

2013 ◽  
Author(s):  
Ju-Yi Yen ◽  
Marc Yor
Keyword(s):  
Brownian Motion ◽  
Local Times ◽  
Excursion Theory

Asymptotic variance parameters for the boundary local times of reflected Brownian motion on a compact interval

1992 ◽  
Vol 29 (04) ◽  
pp. 996-1002 ◽  
Author(s):  
R. J. Williams
Keyword(s):  
Brownian Motion ◽  
Asymptotic Variance ◽  
Hitting Time ◽  
Local Times ◽  
Compact Interval ◽  
Reflected Brownian Motion ◽  
Direct Derivation ◽  
One Dimensional ◽  
Brownian Motion With Drift ◽  
Variance Parameters

A direct derivation is given of a formula for the normalized asymptotic variance parameters of the boundary local times of reflected Brownian motion (with drift) on a compact interval. This formula was previously obtained by Berger and Whitt using an M/M/1/C queue approximation to the reflected Brownian motion. The bivariate Laplace transform of the hitting time of a level and the boundary local time up to that hitting time, for a one-dimensional reflected Brownian motion with drift, is obtained as part of the derivation.

scholarly journals α-time fractional Brownian motion: PDE connections and local times

2012 ◽  
Vol 16 ◽  
pp. 1-24 ◽  
Author(s):  
Erkan Nane ◽  
Dongsheng Wu ◽  
Yimin Xiao
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Local Times

scholarly journals Brownian local times

1995 ◽  
Vol 8 (3) ◽  
pp. 209-232 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Brownian Motion ◽  
Brownian Bridge ◽  
Distribution Functions ◽  
Local Times ◽  
Explicit Formulas ◽  
Brownian Excursion ◽  
Reflecting Brownian Motion ◽  
Brownian Meander

In this paper explicit formulas are given for the distribution functions and the moments of the local times of the Brownian motion, the reflecting Brownian motion, the Brownian meander, the Brownian bridge, the reflecting Brownian bridge and the Brownian excursion.

scholarly journals Some extensions of Pitman and Ray-Knight theorems for penalized Brownian motions and their local times, IV

2007 ◽  
Vol 44 (4) ◽  
pp. 469-516 ◽  
Author(s):  
Bernard Roynette ◽  
Pierre Vallois ◽  
Marc Yor
Keyword(s):  
Brownian Motion ◽  
Local Times ◽  
Brownian Motions ◽  
Pitman’S Theorem

We show that Pitman’s theorem relating Brownian motion and the BES (3) process, as well as the Ray-Knight theorems for Brownian local times remain valid, mutatis mutandis, under the limiting laws of Brownian motion penalized by a function of its one-sided maximum.

scholarly journals The Integral of the Supremum Process of Brownian Motion

2009 ◽  
Vol 46 (2) ◽  
pp. 593-600 ◽  
Author(s):  
Svante Janson ◽  
Niclas Petersson
Keyword(s):  
Brownian Motion ◽  
Laplace Transform ◽  
Explicit Formula ◽  
Local Time ◽  
Time Interval ◽  
Standard Brownian Motion ◽  
Excursion Theory ◽  
Double Laplace Transform ◽  
The Laplace Transform ◽  
Supremum Process

In this paper we study the integral of the supremum process of standard Brownian motion. We present an explicit formula for the moments of the integral (or area)(T) covered by the process in the time interval [0,T]. The Laplace transform of(T) follows as a consequence. The main proof involves a double Laplace transform of(T) and is based on excursion theory and local time for Brownian motion.

scholarly journals Occupation time problem for multifractional Brownian motion

2019 ◽  
Vol 39 (1) ◽  
pp. 99-113
Author(s):  
Mohamed Ait Ouahra ◽  
Raby Guerbaz ◽  
Hanae Ouahhabi ◽  
Aissa Sghir
Keyword(s):  
Brownian Motion ◽  
Sample Path ◽  
Local Limit ◽  
Continuous Functions ◽  
Occupation Time ◽  
Local Times ◽  
Additive Functionals ◽  
Multifractional Brownian Motion ◽  
Sample Path Properties ◽  
Time Problem

In this paper, by using a Fourier analytic approach, we investigate sample path properties of the fractional derivatives of multifractional Brownian motion local times. We also show that those additive functionals satisfy a property of local asymptotic self-similarity. As a consequence, we derive some local limit theorems for the occupation time of multifractional Brownian motion in the space of continuous functions. 

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