Approximation of transition densities of stochastic differential equations by saddlepoint methods applied to small-time Ito–Taylor sample-path expansions

Likelihood-based inference for parameters of stochastic differential equation (SDE) models is challenging because for most SDEs the transition density is unknown. We propose a method for estimating the transition density that involves expanding the sample path as an Ito–Taylor series, calculating the moment generating function of the retained terms in the Ito–Taylor expansion, then employing a saddlepoint approximation. We perform a numerical comparison with two other methods similarly based on small-time expansions and discuss the pros and cons of our new method relative to other approaches.