The last zero-crossing of an iterated brownian motion with drift

AbstractA new approach to the problem of finding the distribution of integral functionals under the excursion measure is presented. It is based on the technique of excursion straddling a time, stochastic analysis, and calculus on local time, and it is done for Brownian motion with drift reflecting at 0, and under some additional assumptions for some class of Itó diffusions. The new method is an alternative to the classical potential-theoretic approach and gives new specific formulas for distributions under the excursion measure.

Download Full-text

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

Journal of Applied Probability ◽

10.1017/s0021900200043850 ◽

1992 ◽

Vol 29 (04) ◽

pp. 996-1002 ◽

Cited By ~ 2

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.

Download Full-text

The ruin problem and cover times of asymmetric random walks and Brownian motions

Advances in Applied Probability ◽

10.1017/s0001867800009836 ◽

2000 ◽

Vol 32 (01) ◽

pp. 177-192 ◽

Cited By ~ 3

Author(s):

K. S. Chong ◽

Richard Cowan ◽

Lars Holst

Keyword(s):

Brownian Motion ◽

Random Walk ◽

Random Walks ◽

Cover Time ◽

Brownian Motions ◽

Brownian Motion With Drift ◽

And Brownian Motion ◽

Cover Times ◽

Ruin Problem

A simple asymmetric random walk on the integers is stopped when its range is of a given length. When and where is it stopped? Analogous questions can be stated for a Brownian motion. Such problems are studied using results for the classical ruin problem, yielding results for the cover time and the range, both for asymmetric random walks and Brownian motion with drift.

Download Full-text

Occupation Times, Drawdowns, and Drawups for One-Dimensional Regular Diffusions

Advances in Applied Probability ◽

10.1239/aap/1427814588 ◽

2015 ◽

Vol 47 (1) ◽

pp. 210-230 ◽

Cited By ~ 18

Author(s):

Hongzhong Zhang

Keyword(s):

Brownian Motion ◽

Diffusion Process ◽

Time Horizon ◽

Three Dimensional ◽

Bessel Processes ◽

Occupation Times ◽

Exponential Time ◽

One Dimensional ◽

Brownian Motion With Drift ◽

Running Maximum

The drawdown process of a one-dimensional regular diffusion process X is given by X reflected at its running maximum. The drawup process is given by X reflected at its running minimum. We calculate the probability that a drawdown precedes a drawup in an exponential time-horizon. We then study the law of the occupation times of the drawdown process and the drawup process. These results are applied to address problems in risk analysis and for option pricing of the drawdown process. Finally, we present examples of Brownian motion with drift and three-dimensional Bessel processes, where we prove an identity in law.

Download Full-text