[1] Gareth O. Roberts, Omiros Papaspiliopoulos, and Petros Dellaportas. Bayesian inference for non-Gaussian Ornstein-Uhlenbeck stochastic volatility processes. J. R. Stat. Soc. Ser. B Stat. Methodol., 66(2):369-393, 2004. [ bib | http ]
We develop Markov chain Monte Carlo methodology for Bayesian inference for non-Gaussian Ornstein-Uhlenbeck stochastic volatility processes. The approach introduced involves expressing the unobserved stochastic volatility process in terms of a suitable marked Poisson process. We introduce two specific classes of Metropolis-Hastings algorithms which correspond to different ways of jointly parameterizing the marked point process and the model parameters. The performance of the methods is investigated for different types of simulated data. The approach is extended to consider the case where the volatility process is expressed as a superposition of Ornstein–Uhlenbeck processes. We apply our methodology to the US dollar-Deutschmark exchange rate.

Keywords: Data augmentation; Lévy processes; Marked point processes; Markov chain Monte Carlo methods; Non-centred parameterizations; Stochastic volatility
[2] Alexandros Beskos, Omiros Papaspiliopoulos, Gareth O. Roberts, and Paul Fearnhead. Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes. J. R. Stat. Soc. Ser. B Stat. Methodol., 68(3):333-382, 2006. With discussions and a reply by the authors. [ bib | http ]
The objective of the paper is to present a novel methodology for likelihood-based inference for discretely observed diffusions. We propose Monte Carlo methods, which build on recent advances on the exact simulation of diffusions, for performing maximum likelihood and Bayesian estimation.

Keywords: Cox-Ingersoll-Ross model; EM algorithm; Graphical models; Markov chain Monte Carlo methods; Monte Carlo maximum likelihood; Retrospective sampling
[3] Alexandros Beskos, Omiros Papaspiliopoulos, and Gareth Roberts. Monte Carlo maximum likelihood estimation for discretely observed diffusion processes. Ann. Statist., 37(1):223-245, 2009. [ bib | http ]
This paper introduces a Monte Carlo method for maximum likelihood inference in the context of discretely observed diffusion processes. The method gives unbiased and a.s. continuous estimators of the likelihood function for a family of diffusion models and its performance in numerical examples is computationally efficient. It uses a recently developed technique for the exact simulation of diffusions, and involves no discretization error. We show that, under regularity conditions, the Monte Carlo MLE converges a.s. to the true MLE. For datasize n ->, we show that the number of Monte Carlo iterations should be tuned as O(n1/2) and we demonstrate the consistency properties of the Monte Carlo MLE as an estimator of the true parameter value.

Keywords: Coupling; uniform convergence; exact simulation; linear diffusion processes; random function; SLLN on Banach space
[4] O. Papaspiliopoulos, G. Roberts, and O. Stramer. Data augmentation for diffusions. Journal of Computational and Graphical Statistics, to appear, 2013. [ bib | .pdf ]
The problem of formal likelihood-based (either classical or Bayesian) inference for discretely observed multi-dimensional diffusions is particularly challenging. In principle this involves data augmentation of the observation data to give representations of the entire diffusion trajectory. Most currently proposed methodology splits broadly into two classes: either through the discretisation of idealised approaches for the continuous-time diffusion setup; or through the use of standard finite-dimensional methodologies discretisation of the diffusion model. The connections between these approaches have not been well-studied. This paper will provide a unified framework bringing together these approaches, demonstrating connections, and in some cases surprising differences. As a result, we provide, for the first time, theoretical justification for the various methods of imputing missing data. The inference problems are particularly challenging for irreducible diffusions, and our framework is correspondingly more complex in that case. Therefore we treat the reducible and irreducible cases differently within the paper. Supplementary materials for the article are available on line.

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