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Bayesian inference for stable Lévy–driven stochastic differential equations with high‐frequency data
Author(s) -
Jasra Ajay,
Kamatani Kengo,
Masuda Hiroki
Publication year - 2019
Publication title -
scandinavian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.359
H-Index - 65
eISSN - 1467-9469
pISSN - 0303-6898
DOI - 10.1111/sjos.12362
Subject(s) - mathematics , markov chain monte carlo , frequentist inference , inference , bayesian inference , bayesian probability , metropolis–hastings algorithm , reversible jump markov chain monte carlo , likelihood function , stochastic differential equation , algorithm , statistics , computer science , artificial intelligence , estimation theory
In this paper, we consider parametric Bayesian inference for stochastic differential equations driven by a pure‐jump stable Lévy process, which is observed at high frequency. In most cases of practical interest, the likelihood function is not available; hence, we use a quasi‐likelihood and place an associated prior on the unknown parameters. It is shown under regularity conditions that there is a Bernstein–von Mises theorem associated to the posterior. We then develop a Markov chain Monte Carlo algorithm for Bayesian inference, and assisted with theoretical results, we show how to scale Metropolis–Hastings proposals when the frequency of the data grows, in order to prevent the acceptance ratio from going to zero in the large data limit. Our algorithm is presented on numerical examples that help verify our theoretical findings.