Research on Hawkes Process for Machine Learning research part10

Convergence of Heavy-Tailed Hawkes Processes and the Microstructure of Rough Volatility

Authors: Ulrich Horst, Wei Xu, Rouyi Zhang

Abstract: We establish the weak convergence of the intensity of a nearly-unstable Hawkes process with heavy-tailed kernel. Our result is used to derive a scaling limit for a financial market model where orders to buy or sell an asset arrive according to a Hawkes process with power-law kernel. After suitable rescaling the price-volatility process converges weakly to a rough Heston model. Our convergence result is much stronger than previously established ones that have either focused on light-tailed kernels or the convergence of integrated volatility process. The key is to establish the tightness of the family of rescaled volatility processes. This is achieved by introducing a new methods to establish the C-tightness of càdlàg processes based on the classical Kolmogorov-Chentsov tightness criterion for continuous processes.