One-dimensional Brownian motion and the three-dimensional Bessel process

1974 ◽  
Vol 6 (02) ◽  
pp. 223-224 ◽  
Author(s):  
J. W. Pitman
Keyword(s):  
Brownian Motion ◽  
Three Dimensional ◽  
Bessel Process ◽  
One Dimensional

One-dimensional Brownian motion and the three-dimensional Bessel process

1975 ◽  
Vol 7 (03) ◽  
pp. 511-526 ◽  
Author(s):  
James W. Pitman
Keyword(s):  
Brownian Motion ◽  
Three Dimensional ◽  
Bessel Process ◽  
Simple Path ◽  
Radial Part ◽  
One Dimensional ◽  
Path Decompositions

A simple path transformation is described which connects one-dimensional Brownian motion with the radial part of three-dimensional Brownian motion. This provides simple proofs of various path decompositions for these processes described by David Williams.

One-dimensional Brownian motion and the three-dimensional Bessel process

1975 ◽  
Vol 7 (3) ◽  
pp. 511-526 ◽  
Author(s):  
James W. Pitman
Keyword(s):  
Brownian Motion ◽  
Three Dimensional ◽  
Bessel Process ◽  
Simple Path ◽  
Radial Part ◽  
One Dimensional ◽  
Path Decompositions

A simple path transformation is described which connects one-dimensional Brownian motion with the radial part of three-dimensional Brownian motion. This provides simple proofs of various path decompositions for these processes described by David Williams.

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.

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.

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.

Quasi-stationary distributions for a Brownian motion with drift and associated limit laws

1994 ◽  
Vol 31 (4) ◽  
pp. 911-920 ◽  
Author(s):  
Servet Martinez ◽  
Jaime San Martin
Keyword(s):  
Brownian Motion ◽  
Limit Distribution ◽  
Invariant Measures ◽  
Three Dimensional ◽  
Transition Density ◽  
Bessel Process ◽  
Limit Laws ◽  
Brownian Motion With Drift ◽  
Infinite Mass ◽  
Negative Drift

We prove that the quasi-invariant measures associated to a Brownian motion with negative drift X form a one-parameter family. The minimal one is a probability measure inducing the transition density of a three-dimensional Bessel process, and it is shown that it is the density of the limit distribution limt→∞Px(X A | τ > t). It is also shown that the minimal quasi-invariant measure of infinite mass induces the density of the limit distribution ) which is the law of a Bessel process with drift.

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

1984 ◽  
Vol 21 (03) ◽  
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.

Quasi-stationary distributions for a Brownian motion with drift and associated limit laws

1994 ◽  
Vol 31 (04) ◽  
pp. 911-920 ◽  
Author(s):  
Servet Martinez ◽  
Jaime San Martin
Keyword(s):  
Brownian Motion ◽  
Limit Distribution ◽  
Invariant Measures ◽  
Three Dimensional ◽  
Transition Density ◽  
Bessel Process ◽  
Limit Laws ◽  
Brownian Motion With Drift ◽  
Infinite Mass ◽  
Negative Drift

We prove that the quasi-invariant measures associated to a Brownian motion with negative drift X form a one-parameter family. The minimal one is a probability measure inducing the transition density of a three-dimensional Bessel process, and it is shown that it is the density of the limit distribution lim t→∞ P x (X A | τ > t). It is also shown that the minimal quasi-invariant measure of infinite mass induces the density of the limit distribution ) which is the law of a Bessel process with drift.

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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