Saddlepoint approximations for continuous-time Markov processes

This paper proposes saddlepoint expansions as a means to generate closed-form approximations to the transition densities and cumulative distribution functions of Markov processes. This method is applicable to a large class of models considered in finance, for which a Laplace or characteristic functions, but not the transition density, can be found in closed form. But even when such a computation is not possible explicitly, we go one step further by showing how useful approximations can be obtained by replacing the Laplace or characteristic functions by an expansion in small time.