Nonparametric Bayesian Estimation of a Hölder Continuous Diffusion Coefficient

Braz. J. Probab. Stat., 34(3): 537–579. 2020.

46 Pages Posted: 27 Dec 2017 Last revised: 22 Jul 2020

See all articles by Shota Gugushvili

Shota Gugushvili

Wageningen University & Research

Frank van der Meulen

Delft University of Technology - Delft Institute of Applied Mathematics (DIAM)

Moritz Schauer

Leiden University

Peter Spreij

Universiteit van Amsterdam, Korteweg-de Vries Institute for Mathematics

Date Written: June 1, 2020

Abstract

We consider a nonparametric Bayesian approach to estimate the diffusion coefficient of a stochastic differential equation given discrete time observations over a fixed time interval. As a prior on the diffusion coefficient, we employ a histogram-type prior with piecewise constant realisations on bins forming a partition of the time interval. Specifically, these constants are realizations of independent inverse Gamma distributed randoma variables. We justify our approach by deriving the rate at which the corresponding posterior distribution asymptotically concentrates around the data-generating diffusion coefficient. This posterior contraction rate turns out to be optimal for estimation of a Hölder-continuous diffusion coefficient with smoothness parameter 0<λ≤1. Our approach is straightforward to implement, as the posterior distributions turn out to be inverse Gamma again, and leads to good practical results in a wide range of simulation examples. Finally, we apply our method on exchange rate data sets.

Keywords: Diffusion Coefficient, Gaussian Likelihood, Non-Parametric Bayesian Estimation, Pseudo-likelihood, Posterior Contraction Rate, Stochastic Differential Equation, Volatility

JEL Classification: C11, C13, C14, C22

Suggested Citation

Gugushvili, Shota and van der Meulen, Frank and Schauer, Moritz and Spreij, Peter, Nonparametric Bayesian Estimation of a Hölder Continuous Diffusion Coefficient (June 1, 2020). Braz. J. Probab. Stat., 34(3): 537–579. 2020., Available at SSRN: https://ssrn.com/abstract=3093100 or http://dx.doi.org/10.2139/ssrn.3093100

Shota Gugushvili (Contact Author)

Wageningen University & Research ( email )

Biometris, WUR
P.O. Box 16
Wageningen, Gelderland 6700 AA
Netherlands

HOME PAGE: http://https://gugushvili.github.io/

Frank Van der Meulen

Delft University of Technology - Delft Institute of Applied Mathematics (DIAM) ( email )

Mekelweg 4
Delft, Holland 2628
Netherlands

Moritz Schauer

Leiden University ( email )

Postbus 9500
Leiden, Zuid Holland 2300 RA
Netherlands

Peter Spreij

Universiteit van Amsterdam, Korteweg-de Vries Institute for Mathematics ( email )

PO Box 94248
Amsterdam, 1090GE
Netherlands
+31 20 5256070 (Phone)

HOME PAGE: http://https://staff.fnwi.uva.nl/p.j.c.spreij/

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