The Doob-McKean identity for stable L\'evy processes

TL;DR

This paper extends the classical Doob--McKean identity from Brownian motion to isotropic -stable processes, establishing a new analogue that broadens understanding of conditioned stable processes.

Contribution

It introduces a natural analogue of the Doob--McKean identity for isotropic -stable processes, generalizing the classical Brownian motion result.

Findings

Established the stable process analogue of the Doob--McKean identity.

Provided a new interpretation of the components in the stable process setting.

Extended classical stochastic identities to a broader class of processes.

Abstract

We re-examine the celebrated Doob--McKean identity that identifies a conditioned one-dimensional Brownian motion as the radial part of a 3-dimensional Brownian motion or, equivalently, a Bessel-3 process, albeit now in the analogous setting of isotropic $\alpha$-stable processes. We find a natural analogue that matches the Brownian setting, with the role of the Brownian motion replaced by that of the isotropic $\alpha$-stable process, providing one interprets the components of the original identity in the right way.