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

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 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 method to establish the ‐tightness of càdlàg processes based on the classical Kolmogorov‐Chentsov tightness criterion for continuous processes.