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

TL;DR

This paper develops simulation schemes for continuous-time ARMA processes driven by tempered stable Lévy noises, extending series representations and providing error bounds, with Monte Carlo experiments demonstrating effectiveness.

Contribution

It introduces a novel series representation for p-tempered α-stable distributions and applies it to simulate CARMA processes with error analysis.

Findings

Derived a series representation for p-tempered α-stable distributions.

Proved approximation error bounds for the simulation method.

Conducted Monte Carlo experiments demonstrating the approach's usefulness.

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.