Paper: arXiv 2312.08784

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 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.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a rigorous theoretical contribution by proving the weak convergence of heavy-tailed Hawkes processes to rough Heston models using a novel C-tightness criterion. While mathematically sophisticated and novel in its limit theorems, it is purely theoretical with no empirical backtesting or real-market data application.

Research Flowchart

  flowchart TD
    A[Research Goal: Link Heavy-Tailed Hawkes Processes to Rough Volatility] --> B[Input: Asset Orders as Heavy-Tailed Hawkes Process]
    B --> C[Method: Apply Kolmogorov-Chentsov C-tightness for Càdlàg Processes]
    C --> D[Process: Rescaling & Weak Convergence Analysis of Intensity]
    D --> E[Outcome: Convergence to Rough Heston Model]
    D --> F[Outcome: Stronger Convergence Bounds than Light-Tailed Models]

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