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Bayesian inference and model selection for multi-dimensional diffusion process models with non-parametric drift and constant diffusivity

Hoh, Tjun Yee; (2019) Bayesian inference and model selection for multi-dimensional diffusion process models with non-parametric drift and constant diffusivity. Doctoral thesis (Ph.D), UCL (University College London). Green open access

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Abstract

For a multi-dimensional, partially observed diffusion process model with unknown drift and variable-independent diffusivity, we construct a composite methodology to perform Bayesian inference for the coefficients. Recent development of non-parametric Bayesian estimation of the drift has been restricted to dimension one, since the local time process is unavailable in the multi-dimensional case. We involve the empirical measure instead and show that the drift likelihood has a quadratic form, which allows a conjugate Gaussian measure prior whose precision operator is chosen to be a high order differential operator. We detail a computationally efficient pseudo-spectral method for solving the posterior mean, and describe how inference for the drift can be constrained to allow only conservative drifts. We also adapt a Langevin MCMC approach to sampling from diffusion bridges as a data augmentation scheme. To sample from the diffusivity, we specify an Inverse Wishart prior and implement a random walk Metropolis-Hastings algorithm. Evaluation of model fit for diffusion processes historically involved frequentist goodness-of-fit testing for fully parametric null models. We extend an existing transition density-based omnibus test to the null model case with non-parametric drift. We study the finite-sample behaviour of the test statistic and show that existing asymptotic results are inappropriate for settings involving real data. We implement the Bayesian discrepancy p-value to complement our inference methodology. With the goal of model improvement in mind, we describe how outlier removal and systematic sub-sampling of the data can be beneficial.

Type: Thesis (Doctoral)
Qualification: Ph.D
Title: Bayesian inference and model selection for multi-dimensional diffusion process models with non-parametric drift and constant diffusivity
Event: UCL (University College London)
Open access status: An open access version is available from UCL Discovery
Language: English
Additional information: Copyright © The Author 2019. Original content in this thesis is licensed under the terms of the Creative Commons Attribution 4.0 International (CC BY 4.0) Licence (https://creativecommons.org/licenses/by/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request.
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Statistical Science
URI: https://discovery.ucl.ac.uk/id/eprint/10076641
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