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Nonparametric Bayesian inference for Gamma-type Lévy subordinators

Journal article
Authors D. Belomestny
S. Gugushvili
Moritz Schauer
P. Spreij
Published in Communications in Mathematical Sciences
Volume 17
Issue 3
Pages 781-816
ISSN 1539-6746
Publication year 2019
Published at Department of Mathematical Sciences, Applied Mathematics and Statistics
Pages 781-816
Language en
Links dx.doi.org/10.4310/CMS.2019.v17.n3....
Keywords Bridge sampling, Data augmentation, Gamma process, Levy process, Levy density, MCMC, convergence-rates, model, Mathematics
Subject categories Mathematics

Abstract

Given discrete time observations over a growing time interval, we consider a nonparametric Bayesian approach to estimation of the Levy density of a Levy process belonging to a flexible class of infinite activity subordinators. Posterior inference is performed via MCMC, and we circumvent the problem of the intractable likelihood via the data augmentation device, that in our case relies on bridge process sampling via Gamma process bridges. Our approach also requires the use of a new infinite-dimensional form of a reversible jump MCMC algorithm. We show that our method leads to good practical results in challenging simulation examples. On the theoretical side, we establish that our nonparametric Bayesian procedure is consistent: in the low frequency data setting, with equispaced in time observations and intervals between successive observations remaining fixed, the posterior asymptotically, as the sample size n ->infinity, concentrates around the Levy density under which the data have been generated. Finally, we test our method on a classical insurance dataset.

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