Density of Generalized Verhulst Process and Bessel Process with Constant Drift

2016 ◽  
Author(s):  
Zhenyu Cui
Keyword(s):  
Bessel Process ◽  
Verhulst Process

A boundary crossing probability for the Bessel process

1998 ◽  
Vol 30 (3) ◽  
pp. 807-830 ◽  
Author(s):  
Rebecca A. Betensky
Keyword(s):  
Nonparametric Test ◽  
Bessel Process ◽  
Boundary Crossing ◽  
Test Statistic ◽  
Boundary Crossing Probability ◽  
Significance Level ◽  
Straight Line ◽  
Crossing Probability ◽  
Nonparametric Test Statistic ◽  
First Crossing Time

Analytic approximations are derived for the distribution of the first crossing time of a straight-line boundary by a d-dimensional Bessel process and its discrete time analogue. The main ingredient for the approximations is the conditional probability that the process crossed the boundary before time m, given its location beneath the boundary at time m. The boundary crossing probability is of interest as the significance level and power of a sequential test comparing d+1 treatments using an O'Brien-Fleming (1979) stopping boundary (see Betensky 1996). Also, it is shown by DeLong (1980) to be the limiting distribution of a nonparametric test statistic for multiple regression. The approximations are compared with exact values from the literature and with values from a Monte Carlo simulation.

scholarly journals Fractionally integrated Bessel process

2021 ◽  
Vol 477 (2250) ◽  
pp. 20200934
Author(s):  
Georgiy Shevchenko ◽  
Dmytro Zatula
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Hölder Continuity ◽  
Bessel Process ◽  
Holder Continuity ◽  
Basic Properties

We consider a fractionally integrated Bessel process defined by Y s δ , H = ∫ 0 ∞ ( u H − ( 1 / 2 ) − ( u − s ) + H − ( 1 / 2 ) ) d X u δ , where X δ is the Bessel process of dimension δ  > 2. We discuss the relation of this process to the fractional Brownian motion at its maximum, study the basic properties of the process and prove its Hölder continuity.

Invariance formulas for stopping times of squared Bessel process

2018 ◽  
Vol 36 (4) ◽  
pp. 671-699
Author(s):  
Jacek Jakubowski ◽  
Maciej Wiśniewolski
Keyword(s):  
Bessel Process ◽  
Stopping Times ◽  
Squared Bessel Process

scholarly journals On conditional Ornstein–Uhlenbeck processes

1984 ◽  
Vol 16 (04) ◽  
pp. 920-922
Author(s):  
P. Salminen
Keyword(s):  
Brownian Motion ◽  
Three Dimensional ◽  
Bessel Process ◽  
Uhlenbeck Process ◽  
Brownian Motions ◽  
Ornstein Uhlenbeck Process ◽  
The Law ◽  
Similar Description

It is well known that the law of a Brownian motion started from x > 0 and conditioned never to hit 0 is identical with the law of a three-dimensional Bessel process started from x. Here we show that a similar description is valid for all linear Ornstein–Uhlenbeck Brownian motions. Further, using the same techniques, it is seen that we may construct a non-stationary Ornstein–Uhlenbeck process from a stationary one.

The first exit time for a Bessel process from the minimum and maximum random domains

2009 ◽  
Vol 79 (20) ◽  
pp. 2115-2123 ◽  
Author(s):  
Lixin Song ◽  
Dawei Lu ◽  
Jinghai Feng
Keyword(s):  
Exit Time ◽  
Bessel Process ◽  
First Exit Time ◽  
Random Domains

High Excursions of Bessel Process and Other Processes of Bessel Type

2019 ◽  
Vol 100 (1) ◽  
pp. 346-348
Author(s):  
V. I. Piterbarg ◽  
I. V. Rodionov
Keyword(s):  
Bessel Process
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