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

Rogers, L. C. G.

Ann. Probab., Tome 9 (1981) no. 6, p. 561-572 / Harvested from Project Euclid

Résumé

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 $X$.

Détail

Publié le : 1981-08-14
Classification:  Brownian motion,  Bessel process,  one dimensional diffusion,  scale function,  speed measure,  2M-X property,  path decomposition,  Markov kernel,  60J25,  60J35,  60J65,  60J60