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.
This work was initiated while the both authors were based in School of Mathematics and Statistics at the University of Sydney, and was partially supported by JSPS Grants-in-Aid for Scientific Research 20K22301 and 21K03347.
The authors are grateful to the anonymous referee for their valuable feedback that helped improve the quality of this manuscript. The opinions expressed by the authors are solely their own and do not reflect those of the Commonwealth Bank of Australia and its related entities.
"Numerical aspects of shot noise representation of infinitely divisible laws and related processes." Probab. Surveys 18 201 - 271, 2021. https://doi.org/10.1214/20-PS359