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

TL;DR

This paper derives explicit transition laws for $p$-tempered $eta$-stable Ornstein-Uhlenbeck processes, enabling simulation and analysis in both univariate and multivariate cases, especially when $p \,\le\, \alpha$.

Contribution

It provides a novel explicit representation of transition laws for $p$-tempered $eta$-stable OU processes, facilitating simulation and extending to multivariate cases.

Findings

Explicit transition law formulas derived for $p$-tempered $eta$-stable OU processes.

Methodology for simulation based on the explicit transition laws.

Special treatment for cases where $p \le \alpha$, handling increased complexity.

Abstract

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 a methodology for simulation. Our results apply in both the univariate and multivariate cases. Special attention is given to the case where $p\le\alpha$, which is more complicated and requires additional care.