Norm-dependent Lamperti-type MAP representations of stable processes and Brownian motions in the orthant

TL;DR

This paper establishes a norm-dependent Lamperti-type representation linking self-similar Markov processes in the orthant with Markov additive processes, providing explicit descriptions for processes involving stable processes and Brownian motions.

Contribution

It introduces a novel norm-dependent Lamperti transform for self-similar Markov processes and characterizes associated MAPs in multidimensional orthant settings.

Findings

Describes MAPs for killed and reflected stable processes in the orthant.

Provides explicit MAP characterizations for reflected Brownian motion.

Connects self-similar processes with Markov additive processes via a new norm-dependent transform.

Abstract

We start by remarking a one-to-one correspondence between self-similar Markov processes (ssMps) on a Banach space and Markov additive processes (MAPs) that is analogous to the well-known one between positive ssMps and L\'evy processes through the renowned Lamperti-transform, with the main difference that ours is norm-dependent. We then consider multidimensional self-similar Markov processes obtained by killing or by reflecting a stable process or Brownian motion in the orthant and we then fully describe the MAPs associated to them using the $L_1$-norm. Namely, we describe the MAP underlying the ssMp obtained by killing a $d$-dimensional $\alpha$-stable process when it leaves the orthant and the one obtained by reflecting it back in the orthant continuously (or by a jump); finally, we also describe the MAP underlying $d$-dimensional Brownian motion reflected in the orthant. The first three of the aforementioned examples are pure-jump, and the last is a diffusion, so their characterization is given through their L\'evy system, generator and/or through the modulated SDE that defines them, respectively.