Journal of Applied Mathematics and Stochastic Analysis
Volume 2007 (2007), Article ID 92723, 19 pages
doi:10.1155/2007/92723
Research Article
A Family of Non-Gaussian Martingales with Gaussian Marginals
School of Mathematical Sciences, Monash University, Clayton VIC 3800, Australia
Received 16 February 2007; Revised 22 May 2007; Accepted 10 June 2007
Copyright © 2007 Kais Hamza and Fima C. Klebaner. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We construct a family of martingales with Gaussian marginal distributions. We give a weak construction as Markov, inhomogeneous in time processes, and compute their infinitesimal generators. We give the predictable quadratic variation and show that the paths are not continuous. The construction uses distributions Gσ having a log-convolution semigroup property. Further, we categorize these
processes as belonging to one of two classes, one of which is made up of piecewise deterministic pure jump processes. This class includes the case where Gσ is an inverse log-Poisson distribution. The processes in the second
class include the case where Gσ is an inverse log-gamma distribution. The richness of the family has the potential to allow for the imposition of specifications other than the marginal distributions.