Diffusion-type models with given marginal distribution and autocorrelation function

Flexible stationary diffusion-type models are developed that can fit both the marginal distribution and the correlation structure found in many time series from, for example, finance and turbulence. Diffusion models with linear drift and a known and prespecified marginal distribution are studied, and the diffusion coefficients corresponding to a large number of common probability distributions are found explicitly. An approximation to the diffusion coefficient based on saddlepoint techniques is developed for use in cases where there is no explicit expression for the diffusion coefficient. It is demonstrated theoretically as well as in a study of turbulence data that sums of diffusions with linear drift can fit complex correlation structures. Any infinitely divisible distribution satisfying a weak regularity condition can be obtained as a marginal distribution.