Efficient simulation of p-tempered $$\alpha $$-stable OU processes

We develop efficient methods for simulating processes of Ornstein–Uhlenbeck type related to the class p-tempered $$\alpha $$ -stable ( $$\textrm{TS}^p_\alpha ) distributions. Our results hold both univariate and multivariate cases we consider case where distribution is stationary law it background driving Lévy process. In latter case, also derive an explicit representation transition as this was previous known only in certain special $$p=1$$ \in [0,1)$$ . Simulation suggest that our work well practice.