scholarly journals ONE DIMENSIONAL BROWNIAN MOTION WITH HOLDING AND JUMPING BOUNDARY

2022 ◽  
Author(s):  
Yuk Leung
Keyword(s):  
Brownian Motion ◽  
Diffusion Process ◽  
Probability Density ◽  
Boundary Problem ◽  
Exponential Distribution ◽  
Boundary Point ◽  
Nonlocal Boundary ◽  
Unit Interval ◽  
One Dimensional ◽  
Nonlocal Boundary Problem

Let a particle start at some point in the unit interval I := [0, 1] and undergo Brownian motion in I until it hits one of the end points. At this instant the particle stays put for a finite holding time with an exponential distribution and then jumps back to a point inside I with a probability density μ0 or μ1 parametrized by the boundary point it was from. The process starts afresh. The same evolution repeats independently each time. Many probabilistic aspects of this diffusion process are investigated in the paper [10]. The authors in the cited paper call this process diffusion with holding and jumping (DHJ). Our simple aim in this paper is to analyze the eigenvalues of a nonlocal boundary problem arising from this process.

Transition probability density of a certain diffusion process concentrated on a finite spatial interval

1992 ◽  
Vol 29 (2) ◽  
pp. 334-342
Author(s):  
A. Milian
Keyword(s):  
Diffusion Process ◽  
Probability Density ◽  
Transition Probability ◽  
Transition Probability Density ◽  
One Dimensional ◽  
Spatial Interval

We show that under some assumptions a diffusion process satisfying a one-dimensional Itô's equation has a transition probability density concentrated on a finite spatial interval. We give a formula for this density.

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

2015 ◽  
Vol 47 (1) ◽  
pp. 210-230 ◽  
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.

Transition probability density of a certain diffusion process concentrated on a finite spatial interval

1992 ◽  
Vol 29 (02) ◽  
pp. 334-342
Author(s):  
A. Milian
Keyword(s):  
Diffusion Process ◽  
Probability Density ◽  
Transition Probability ◽  
Transition Probability Density ◽  
One Dimensional ◽  
Spatial Interval

We show that under some assumptions a diffusion process satisfying a one-dimensional Itô's equation has a transition probability density concentrated on a finite spatial interval. We give a formula for this density.

scholarly journals The flashing Brownian ratchet and Parrondo’s paradox

2018 ◽  
Vol 5 (1) ◽  
pp. 171685 ◽  
Author(s):  
S. N. Ethier ◽  
Jiyeon Lee
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Diffusion Process ◽  
Directed Motion ◽  
Time And Space ◽  
Brownian Ratchet ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Games Of Chance ◽  
Parrondo’S Paradox

A Brownian ratchet is a one-dimensional diffusion process that drifts towards a minimum of a periodic asymmetric sawtooth potential. A flashing Brownian ratchet is a process that alternates between two regimes, a one-dimensional Brownian motion and a Brownian ratchet, producing directed motion. These processes have been of interest to physicists and biologists for nearly 25 years. The flashing Brownian ratchet is the process that motivated Parrondo’s paradox, in which two fair games of chance, when alternated, produce a winning game. Parrondo’s games are relatively simple, being discrete in time and space. The flashing Brownian ratchet is rather more complicated. We show how one can study the latter process numerically using a random walk approximation.

scholarly journals An analogue of Pitman’s 2M – X theorem for exponential Wiener functionals: Part I: A time-inversion approach

2000 ◽  
Vol 159 ◽  
pp. 125-166 ◽  
Author(s):  
Hiroyuki Matsumoto ◽  
Marc Yor
Keyword(s):  
Brownian Motion ◽  
Diffusion Process ◽  
Infinitesimal Generator ◽  
Time Inversion ◽  
Bessel Processes ◽  
One Dimensional

Let be a one-dimensional Brownian motion with constant drift µ ∈ R starting from 0. In this paper we show thatgives rise to a diffusion process and we explain how this result may be considered as an extension of the celebrated Pitman’s 2M - X theorem. We also derive the infinitesimal generator and some properties of the diffusion process and, in particular, its relation to the generalized Bessel processes.

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

2015 ◽  
Vol 47 (01) ◽  
pp. 210-230 ◽  
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 processXis given byXreflected at its running maximum. The drawup process is given byXreflected 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.

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