Simulating Continuous-Time Autoregressive Moving Average Processes Driven By p-Tempered {\alpha}-Stable L\'evy Processes

Abstract

We discuss simulation schemes for continuous-time autoregressive moving average (CARMA) processes driven by tempered stable L\'evy noises. CARMA processes are the continuous-time analogue of ARMA processes as well as a generalization of Ornstein-Uhlenbeck processes. However, unlike Ornstein-Uhlenbeck processes with a tempered stable driver (see, e.g., Qu et al. (2021)) exact transition probabilities for higher order CARMA processes are not explicitly given. Therefore, we follow the sample path generation method of Kawai (2017) and approximate the driving tempered stable L\'evy process by a truncated series representations. We derive a result of a series representation for ptempered {\alpha}-stable distributions extending Rosi\'nski (2007). We prove approximation error bounds and conduct Monte Carlo experiments to illustrate the usefulness of the approach.