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

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, Lévy processes, infinitely divisible processes and fields and Lévy-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.

37 pages, 8 figures

References in corpus (3)

• Photon Shot Noise Dephasing in the Strong-Dispersive Limit of Circuit QED
• Extremal behavior of stochastic integrals driven by regularly varying Lévy processes
• A general approach to sample path generation of infinitely divisible processes via shot noise representation

Cited by in corpus (4)

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• A recursive representation for decoupling time-state dependent jumps from jump-diffusion processes
• A general approach to sample path generation of infinitely divisible processes via shot noise representation