A general approach to sample path generation of infinitely divisible processes via shot noise representation

TL;DR

This paper presents a unified shot noise-based method for generating sample paths of multivariate infinitely divisible processes, decomposing them into components for efficient approximation and error analysis.

Contribution

It introduces a novel, general approach for simulating infinitely divisible processes using shot noise representation, with detailed error analysis and convergence conditions.

Findings

Approximation based on large jumps is simpler than Gaussian simulation.

Error terms vanish under specific technical conditions.

Scaled small jumps converge to a Gaussian process, improving accuracy.

Abstract

We establish a sample path generation scheme in a unified manner for general multivariate infinitely divisible processes based on shot noise representation of their integrators. The approximation is derived from the decomposition of the infinitely divisible process to three independent components based on jump sizes and timings: the large jumps over a compact time interval, small jumps over the entire time interval and large jumps over an unbounded time interval. The first component is taken as the approximation and is much simpler than simulation of general Gaussian processes, while the latter two components are analyzed as the error. We derive technical conditions for the two error terms to vanish in the limit and for the scaled component on small jumps to converge to a Gaussian process so as to enhance the accuracy of the weak approximation. We provide an extensive collection of examples to highlight the wide practicality of the proposed approach.