Nonparametric Bayesian inference for Gamma-type Lévy subordinators
Journal article, 2019

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.

Data augmentation

Posterior consistency

Levy process

Bridge sampling

Metropolis-Hastings algorithm


Nonparametric Bayesian estimation

Gamma process


Levy density


Reversible jump MCMC


Denis Belomestny

University of Duisburg-Essen

National Research University

Shota Gugushvili

Wageningen University and Research

Moritz Schauer

University of Gothenburg

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Peter Spreij

University of Amsterdam

Radboud University

Communications in Mathematical Sciences

1539-6746 (ISSN)

Vol. 17 3 781-816

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