Gamma-related Ornstein–Uhlenbeck processes and their simulation*

We investigate the distributional properties of two Lévy-driven Ornstein–Uhlenbeck (OU) processes whose stationary distributions are the gamma law and the bilateral gamma law, respectively. The said distributions turn out to be related to the self-decomposable gamma and bilateral gamma laws, and their densities and characteristic functions are here given in closed form. Algorithms for the exact generation of such processes are accordingly derived with the advantage of being significantly faster than those available in the literature and therefore suitable for real-time simulations.