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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 …