In this article we examine two relatively new MCMC methods which
allow for Bayesian inference in di®usion 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 arbi-
trarily \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 di®usion
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 di®usion volatility. The e±cacy 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.