Projects per year
Abstract
We construct a family of selfsimilar Markov martingales with given marginal distributions. This construction uses the selfsimilarity 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 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 transitionrandomising approach and the timechange 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 tκVtκV for some symmetric random variable VV. At the end, we see how we can mimic certain Brownian martingales which are nonMarkovian.
Original language  English 

Pages (fromto)  13411360 
Number of pages  20 
Journal  Bernoulli 
Volume  21 
Issue number  3 
DOIs  
Publication status  Published  2015 
Keywords
 Lévy processes
 martingales with given marginals
 selfsimilar
Projects
 1 Finished

New Stochastic Processes with Applications in Finance
Klebaner, F., Buchmann, B. & Hamza, K.
Australian Research Council (ARC), Monash University
31/07/09 → 31/12/13
Project: Research