Data augmentation for diffusions

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 reducible 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 avilable on line. 1 Overview of likelihood-based inference for diffusions Diffusion processes have gained much popularity as statistical models for observed and latent processes. Among others, their appeal lies in their flexibility to deal with nonlinearity, time-inhomogeneity and heteroscedasticity by specifying two interpretable functionals, their amenability to efficient computations due to their Markov property, and the rich existing mathematical theory about their properties. As a result, they are used as models throughout Science; some book references related with this approach to modeling include Section 5.3 of [1] for physical systems, Section 8.3.3 (in conjunction with Section 6.3) of [12] for systems biology and mass action stochastic kinetics, and Chapter 10 of [27] for interest rates. A mathematically precise specification of a d-dimensional diffusion process V is as the solution of a stochastic differential equation (SDE) of the type: dVs = b(s, Vs; θ1) ds+ σ(s, Vs; θ2) dBs, s ∈ [0, T ] ; (1) where B is an m-dimensional standard Brownian motion, b(·, · ; · ) : R+ ×Rd ×Θ1 → R is the drift and σ(·, · ; · ) : R+ × R × Θ2 → R is the diffusion coefficient. These ICREA and Department of Economics, Universitat Pompeu Fabra, [email protected] Department of Statistics, University of Warwick Department of Statistics and Actuarial Science, University of Iowa, Iowa City, Iowa