Sample Path Generation of Lévy-Driven Continuous-Time Autoregressive Moving Average Processes

Abstract

We address the issue of sample path simulation of Lévy-driven continuous-time autoregressive moving average (CARMA) processes. Approximate discrete-time simulation schemes are constructed along with quantifiable error analysis for stable, second-order and non-negative CARMA processes, based upon the so-called series representation of infinitely divisible laws and associated Lévy processes. We prove that under suitable conditions, the simulation scheme can be improved in terms of second-order structure, finite dimensional laws as well as sample path properties. The simulation procedure is often quite simple and allows one to conduct super-sampling without running the algorithm once again. The computational complexity of the proposed scheme is not affected much by the sampling scheme, such as sampling frequency and irregular spacing. Numerical results are presented throughout to illustrate the effectiveness of the proposed simulation scheme.

Stable Process Approach to Analysis of Systems Under Heavy-Tailed Noise: Modeling and Stochastic Linearization

“…This result implies that the suitable sample number is difficult to determine. Actually, even for such linear processes, their effective Monte Carlo simulation requires sophisticated techniques when n > 1 [30].…”

“…Then, it is reasonable to approximate sat d j (y j ) by linear gain k j = min 1, d j γ α σ y j (30) by Theorem 3. Conversely, once each saturation sat d j (y j ) is approximated by a linear gain k j , the stationary distribution of y j is given in the form of (29) with…”

Numerical aspects of shot noise representation of infinitely divisible laws and related processes

“…We provide the shot noise representation of the stable law obtained via the inverse Lévy measure method [70], which is crucial from a theoretical perspective [26,47,93,94] as well as for practical use [53,54,68], to name a few examples. Example 3.3 (Inverse Lévy measure method for stable random vector).…”

“…We now turn to a numerical scheme for generating approximate sample paths of Lévy-driven CARMA processes via Poisson truncation of shot noise representation [54]. Lévy-driven CARMA processes naturally generalise Gaussian CARMA processes so as to capture asymmetry and heavy tails in a variety of physical and social science settings.…”

“…As one would expect, the widespread use of the numerical technique demands analyses of the associated truncation error. To give some examples of particular stochastic processes, error analyses have been performed for the stable process [12], gamma process [49], tempered stable process [45], higher order fractional stable motion [53] and Lévy-driven CARMA processes [54]. More general treatments of error analysis have been studied, for example, in terms of moments [46] and Gaussian approximation [3,22].…”