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We derive an explicit representation for the transition law of a $p$-tempered$\alpha$-stable process of Ornstein-Uhlenbeck-type and use it to develop amethodology for simulation. Our results apply in both the univariate andmultivariate cases. Special attention is given to the case where $p\le\alpha$,which is more complicated and requires additional care.

On the Transition Laws of $p$-Tempered $\alpha$-Stable OU-Processes