Communications in Mathematical Sciences

Volume 17 (2019)

Number 3

Nonparametric Bayesian inference for Gamma-type Lévy subordinators

Pages: 781 – 816

DOI: http://dx.doi.org/10.4310/CMS.2019.v17.n3.a8

Authors

Denis Belomestny (Faculty of Mathematics, Duisburg-Essen University, Essen, Germany; and National Research University, Higher School of Economics, Moscow, Russia)

Shota Gugushvili (Biometris, Wageningen University and Research, Wageningen, The Netherlands)

Moritz Schauer (Department of Mathematical Sciences, Chalmers University of Technology, Gothenburg, Sweden; and University of Gothenburg, Sweden)

Peter Spreij (Korteweg-de Vries Institute for Mathematics, University of Amsterdam, The Netherlands; and Institute for Mathematics, Astrophysics and Particle Physics, Radboud University, Nijmegen, The Netherlands)

Abstract

Given discrete time observations over a growing time interval, we consider a nonparametric Bayesian approach to estimation of the Lévy density of a Lévy 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 $n \to \infty$, concentrates around the Lévy density under which the data have been generated. Finally, we test our method on a classical insurance dataset.

Keywords

bridge sampling, data augmentation, Gamma process, Lévy process, Lévy density, MCMC, Metropolis-Hastings algorithm, nonparametric Bayesian estimation, posterior consistency, reversible jump MCMC, subordinator, $\theta$-subordinator

2010 Mathematics Subject Classification

Primary 62G20. Secondary 62M30.

The research leading to the results in this paper has received funding from the European Research Council under ERC Grant Agreement 320637.

The research of the first author was supported by the Russian Academic Excellence Project “5-100” and the German Science Foundation research grant (DFG Sachbeihilfe) 406700014.

Received 30 April 2018

Accepted 30 January 2019

Published 30 August 2019