The Annals of Probability

Characterizing all Diffusions with the $2M - X$ Property

L. C. G. Rogers

Full-text: Open access

Abstract

If $(X_t)_{t \geq 0}$ is a Brownian motion on the real line, started at zero, if $M_t \equiv \max\{X_s; s \leq t\}$ and if $Y_t \equiv 2M_t - X_t$ for $t \geq 0$, then $(Y_t)_{t \geq 0}$ is a homogeneous strong Markov process equal in law to the radial part of Brownian motion in three dimensions. This result was discovered by Pitman, and recently Rogers and Pitman have found other one-dimensional diffusions $X$ for which $2M - X$ is again a diffusion. This paper gives a complete characterisation of all such diffusions $X$.

Article information

Source
Ann. Probab., Volume 9, Number 4 (1981), 561-572.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176994362

Digital Object Identifier
doi:10.1214/aop/1176994362

Mathematical Reviews number (MathSciNet)
MR624683

Zentralblatt MATH identifier
0465.60063

JSTOR
links.jstor.org

Subjects
Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60J65: Brownian motion [See also 58J65] 60J60: Diffusion processes [See also 58J65]

Keywords
Brownian motion Bessel process one dimensional diffusion scale function speed measure 2M-X property path decomposition Markov kernel

Citation

Rogers, L. C. G. Characterizing all Diffusions with the $2M - X$ Property. Ann. Probab. 9 (1981), no. 4, 561--572. doi:10.1214/aop/1176994362. https://projecteuclid.org/euclid.aop/1176994362


Export citation