Mimicking self-similar processes

TL;DR

This paper develops methods to construct self-similar Markov martingales with specified marginals, extending classical processes like Brownian motion and Bessel processes, and explores their properties and applications.

Contribution

It introduces two novel approaches for constructing self-similar Markov martingales with given marginals, including examples and analysis of their properties.

Findings

Constructed families of self-similar Markov martingales with prescribed marginals.

Derived infinitesimal generators and path properties of the constructed processes.

Provided examples including Gaussian, Bessel, and stable Lévy martingales.

Abstract

We construct a family of self-similar Markov martingales with given marginal distributions. This construction uses the self-similarity and Markov property of a reference process to produce a family of Markov processes that possess the same marginal distributions as the original process. The resulting processes are also self-similar with the same exponent as the original process. They can be chosen to be martingales under certain conditions. In this paper, we present two approaches to this construction, the transition-randomising approach and the time-change approach. We then compute the infinitesimal generators and obtain some path properties of the resulting processes. We also give some examples, including continuous Gaussian martingales as a generalization of Brownian motion, martingales of the squared Bessel process, stable L\'{e}vy processes as well as an example of an artificial process having the marginals of $t^{\kappa}V$ for some symmetric random variable $V$. At the end, we see how we can mimic certain Brownian martingales which are non-Markovian.