Mimicking Selfsimilar Processes
jeudi 21 juin 2012, 16h00 - 17h00
We construct a family of selfsimilar Markov martingales with given marginal distributions. This construction uses the selfsimilar and Markov properties 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 selfsimilar 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, martingales of the squared Bessel process, stable Lévy processes as well as an example of an artificial process having the marginals of $t^kV$ for some symmetric random variable $V$. We conclude by showing how to mimic a certain family of Brownian martingales and also extend
the construction to non-Markovian continuous martingales.