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We derive an explicit representation for the transition law of a p -tempered $$\alpha $$ α -stable process of Ornstein–Uhlenbeck-type and use it to develop a methodology for simulation. Our results apply in both the univariate and multivariate cases. Special attention is given to the case where $$p\le \alpha $$ p ≤ α , which is more complicated and requires developing the new class of so-called incomplete gamma distributions.

On the transition laws of p-tempered $$\alpha $$-stable OU-processes