In this article we examine two relatively new MCMC methods which allow for Bayesian inference in diffusion models. First, the Monte Carlo within Metropolis (MCWM) algorithm (O'Neil et al. 2000) uses an importance sampling approximation for the likelihood and yields a Markov chain. Our simulation study shows that there exists a limiting stationary distribution that can be made arbitrarily "close" to the posterior distribution (MCWM is not a standard Metropolis-Hastings algorithm, however). The second method, described in Beaumont ( 2003) and generalized in Andrieu and Roberts ( 2009), introduces auxiliary variables and utilizes a standard Metropolis-Hastings algorithm on the enlarged space; this method preserves the original posterior distribution. When applied to diffusion models, this pseudo-marginal (PM) approach can be viewed as a generalization of the popular data augmentation schemes that sample jointly from the missing paths and the parameters of the diffusion volatility. The efficacy of the PM approach is demonstrated in a simulation study of the Cox-Ingersoll-Ross (CIR) and Heston models, and is applied to two well known datasets. Comparisons are made with the MCWM algorithm and the Golightly and Wilkinson (2008) approach.

This article focuses on two methods to approximate the log-likelihood of discretely observed univariate diffusions: (1) the simulation approach using a modified Brownian bridge as the importance sampler, and (2) the closed-form approximation approach. For the case of constant volatility, we give a theoretical justification of the modified Brownian bridge sampler by showing that it is exactly a Brownian bridge. We also discuss computational issues in the simulation approach such as accelerating the numerical variance stabilizing transformation, computing derivatives of the simulated log-likelihood, and choosing initial values of parameter estimates. The two approaches are compared in the context of financial applications under a benchmark model which has an unknown transition density and has no analytical variance stabilizing transformation. The closed-form approximation, particularly the second-order closed-form, is found to be computationally efficient and very accurate when the observation frequency is monthly or higher. It is more accurate in the center than in the tails of the transition density. The simulation approach combined with the variance stabilizing transformation is found to be more reliable than the closed-form approximation when the observation frequency is lower. Both methods perform better when the volatility level is lower, but the simulation method is more robust to the volatility level. When applied to two well-known datasets of daily observations, the two methods yield similar parameter estimates in both datasets but slightly different log-likelihoods in the case of higher volatility.

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. Overview of likelihood-based inference for diffusionsDiffusion 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. where B is an m-dimensional standard Brownian motion, b(·, · ; · ) :is the drift and σ(·, · ; · ) : R + × R d × Θ 2 → R d×m is the diffusion coefficient. These

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