Diffusion processes play a major role in continuous-time modeling in economics, particularly in continuous-time finance. In most cases, however, the transition density function of a diffusion process is not available in closed form. Using Feynman-Kac integration, we construct a recursive scheme for the Laplace transform (in time) of the transition density function. This provides a semianalytic and highly accurate solution to a wide range of asset pricing problems. Generalizations of our technique to functionals of non-Gaussian processes are also briefly discussed.