  Diffusion processes are widely used for modelling real-world phenomena.
Except for select cases however, analytical expressions do not exist for a
diffusion process' transitional probabilities. It is proposed that the cumulant
truncation procedure can be applied to predict the evolution of the cumulants
of the system. These predictions may be subsequently used within the
saddlepoint procedure to approximate the transitional probabilities. An
approximation to the likelihood of the diffusion system is then easily derived.
The method is applicable for a wide-range of diffusion systems - including
multivariate, irreducible diffusion systems that existing estimation schemes
struggle with. Not only is the accuracy of the saddlepoint comparable with the
Hermite expansion - a popular approximation to a diffusion system's
transitional density - it also appears to be less susceptible to increasing
lags between successive samplings of the diffusion process. Furthermore, the
saddlepoint is more stable in regions of the parameter space that are far from
the maximum likelihood estimates. Hence, the saddlepoint method can be
naturally incorporated within a Markov Chain Monte Carlo (MCMC) routine in
order to provide reliable estimates and credibility intervals of the diffusion
model's parameters. The method is applied to fit the Heston model to daily
observations of the S&P 500 and VIX indices from December 2009 to November
2010.
