Tempered stable Ornstein– Uhlenbeck processes: A practical view

We study the one-dimensional Ornstein–Uhlenbeck (OU) processes with marginal law given by tempered stable and tempered infinitely divisible distributions. We investigate the transition law between consecutive observations of these processes and evaluate the characteristic function of integrated tempered OU processes with a view toward practical applications. We then analyze how to draw a random sample from this class of processes by considering both the classical inverse transform algorithm and an acceptance–rejection method based on simulating a stable random sample. Using a maximum likelihood estimation method based on the fast Fourier transform, we empirically assess the simulation algorithm performance.