Non-Gaussian quasi-likelihood estimation of SDE driven by locally stable Lévy process

Research output: Contribution to journalArticle

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Abstract

We address estimation of parametric coefficients of a pure-jump Lévy driven univariate stochastic differential equation (SDE) model, which is observed at high frequency over a fixed time period. It is known from the previous study (Masuda, 2013) that adopting the conventional Gaussian quasi-maximum likelihood estimator then leads to an inconsistent estimator. In this paper, under the assumption that the driving Lévy process is locally stable, we extend the Gaussian framework into a non-Gaussian counterpart, by introducing a novel quasi-likelihood function formally based on the small-time stable approximation of the unknown transition density. The resulting estimator turns out to be asymptotically mixed normally distributed without ergodicity and finite moments for a wide range of the driving pure-jump Lévy processes, showing much better theoretical performance compared with the Gaussian quasi-maximum likelihood estimator. Extensive simulations are carried out to show good estimation accuracy. The case of large-time asymptotics under ergodicity is briefly mentioned as well, where we can deduce an analogous asymptotic normality result.

Original languageEnglish
Pages (from-to)1013-1059
Number of pages47
JournalStochastic Processes and their Applications
Volume129
Issue number3
DOIs
Publication statusPublished - Mar 1 2019

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Quasi-likelihood
Stable Process
Maximum likelihood
Stochastic Equations
Quasi-maximum Likelihood
Differential equations
Ergodicity
Differential equation
Maximum Likelihood Estimator
Estimator
Large Time Asymptotics
Transition Density
Jump Process
Likelihood Function
Asymptotic Normality
Inconsistent
Univariate
Deduce
Jump
Moment

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Modelling and Simulation
  • Applied Mathematics

Cite this

Non-Gaussian quasi-likelihood estimation of SDE driven by locally stable Lévy process. / Masuda, Hiroki.

In: Stochastic Processes and their Applications, Vol. 129, No. 3, 01.03.2019, p. 1013-1059.

Research output: Contribution to journalArticle

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