## Bernoulli

• Bernoulli
• Volume 21, Number 3 (2015), 1341-1360.

### Mimicking self-similar processes

#### 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 Lévy 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.

#### Article information

Source
Bernoulli, Volume 21, Number 3 (2015), 1341-1360.

Dates
Revised: August 2013
First available in Project Euclid: 27 May 2015

https://projecteuclid.org/euclid.bj/1432732022

Digital Object Identifier
doi:10.3150/13-BEJ588

Mathematical Reviews number (MathSciNet)
MR3352046

Zentralblatt MATH identifier
1372.60053

#### Citation

Fan, Jie Yen; Hamza, Kais; Klebaner, Fima. Mimicking self-similar processes. Bernoulli 21 (2015), no. 3, 1341--1360. doi:10.3150/13-BEJ588. https://projecteuclid.org/euclid.bj/1432732022

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