General self-similarity properties for Markov processes and exponential functionals of L{\'e}vy processes

DOI10.1007/S10959-021-01097-2arXiv1807.01878WikidataQ115603644 ScholiaQ115603644MaRDI QIDQ6303872

Publication date: 5 July 2018

Abstract: Positive self-similar Markov processes (pssMp) are positive Markov processes that satisfy the scaling property and it is known that they can be represented as the exponential of a time-changed L'evy process via Lamperti representation. In this work, we are interested in the following problem: what happens if we consider Markov processes in dimension

1

or

2

that satisfy self-similarity properties of a more general form than a scaling property ? Can they all be represented as a function of a time-changed L'evy process ? If not, how can Lamperti representation be generalized ? We show that, not surprisingly, a Markovian process in dimension

1

that satisfies self-similarity properties of a general form can indeed be represented as a function of a time-changed L'evy process, which shows some kind of universality for the classical Lamperti representation in dimension

1

. However, and this is our main result, we show that a Markovian process in dimension

2

that satisfies self-similarity properties of a general form is represented as a function of a time-changed exponential functional of a bivariate L'evy process, and processes that can be represented as a function of a time-changed L'evy process form a strict subclass. This shows that the classical Lamperti representation is not universal in dimension

2

. We briefly discuss the complications that occur in higher dimensions. In dimension

2

we present an example, built from a self-similar fragmentation process, where our representation in term of an exponential functional of a bivariate L'evy process appears naturally and has a nice interpretation in term of the self-similar fragmentation process.

Mathematics Subject Classification ID

Processes with independent increments; Lévy processes (60G51) Continuous-time Markov processes on general state spaces (60J25) Self-similar stochastic processes (60G18)

This page was built for publication: General self-similarity properties for Markov processes and exponential functionals of L{\'e}vy processes

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6303872)