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

TL;DR

This paper reviews shot noise methods for sampling and simulating infinitely divisible laws and processes, emphasizing their practicality in multidimensional settings and providing implementation guidance and numerical examples.

Contribution

It offers a comprehensive overview of shot noise representations for infinitely divisible laws, including truncation techniques and simulation recipes, with practical implementation insights.

Findings

Shot noise methods are effective for multidimensional sampling.

Truncation of series representations enables practical simulation.

Numerical illustrations demonstrate the methods' applicability.

Abstract

The ever-growing appearance of infinitely divisible laws and related processes in various areas, such as physics, mathematical biology, finance and economics, has fuelled an increasing demand for numerical methods of sampling and sample path generation. In this survey, we review shot noise representation with a view towards sampling infinitely divisible laws and generating sample paths of related processes. In contrast to many conventional methods, the shot noise approach remains practical even in the multidimensional setting. We provide a brief introduction to shot noise representations of infinitely divisible laws and related processes, and discuss the truncation of such series representations towards the simulation of infinitely divisible random vectors, L\'evy processes, infinitely divisible processes and fields and L\'evy-driven stochastic differential equations. Essential notions and results towards practical implementation are outlined, and summaries of simulation recipes are provided throughout along with numerical illustrations. Some future research directions are highlighted.