Likelihood based inference for diffusion driven models

Siddhartha Chib: Olin School of Business, Washington University, St Louis, USA chib@wustl.edu

Michael K Pitt: Department of Economics, University of Warwick, Coventry CV4 7AL, UK m.k.pitt@warwick.ac.uk

Neil Shephard: Nuffield College, University of Oxford, Oxford OX1 1NF, UK neil.shephard@nuf.ox.ac.uk

Abstract

This paper provides methods for carrying out likelihood based inference for diffusion driven models, for example discretely observed multivariate diffusions, continuous time stochastic volatility models and counting process models. The diffusions can potentially be non-stationary. Although our methods are sampling based, making use of Markov chain Monte Carlo methods to sample the posterior distribution of the relevant unknowns, our general strategies and details are different from previous work along these lines. The methods we develop are simple to implement and simulation efficient. Importantly, unlike previous methods, the performance of our technique is not worsened, in fact it improves, as the degree of latent augmentation is increased to reduce the bias of the Euler approximation. In addition, our method is not subject to a degeneracy that afflicts previous techniques when the degree of latent augmentation is increased. We also discuss issues of model choice, model checking and filtering. The techniques and ideas are applied to both simulated and real data.

Keywords: Bayes estimation, Brownian bridge, Non-linear diffusion,
Euler approximation, Markov chain Monte Carlo, Metropolis-Hastings
algorithm, Missing data, Simulation, Stochastic differential equation.

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