Paper: arXiv 2605.03703

Authors: Sohaib El Karmi

Abstract

We prove a functional limit theorem for a pair of nearly unstable Hawkes processes coupled through a triangular cross-excitation mechanism, when the two kernels have distinct heavy-tail exponents. This heterogeneous regime produces two different degrees of roughness and, to the best of our knowledge, had not previously been treated in the multivariate nearly unstable setting. As the system approaches criticality, the renormalized intensity processes converge weakly to the unique solution of a coupled stochastic Volterra system driven by two independent Brownian motions. The first component evolves autonomously as a rough fractional diffusion, while the second is driven both by its own noise and by the first component through a convolution cross-kernel. This kernel, expressed as the convolution of the two associated Mittag-Leffler kernels, encodes both roughness exponents and distinguishes the limit from independent univariate limits or classical bivariate Brownian models. We also derive a short-time decorrelation result showing that the functional correlation between the two limiting components vanishes at an explicit polynomial rate governed by the rougher component. Finally, we show that the scale-matching assumption is not structural: without it, the limiting cross-kernel is replaced by an explicitly time-rescaled convolution kernel. The proof combines kernel convergence, tightness, martingale identification via Rebolledo’s theorem, and uniqueness for affine stochastic Volterra equations.

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 breakthrough in probability theory and quantitative finance by deriving functional limit theorems for heterogeneous, nearly-unstable bivariate Hawkes processes. While mathematically sophisticated and highly novel in its theoretical formulation of rough volatility, it is purely theoretical with no empirical backtesting or real-world market data validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Limit theorems for bivariate nearly-unstable Hawkes processes with heterogeneous heavy-tail kernels] --> B[Inputs: Bivariate Hawkes Process with triangular cross-excitation & distinct memory kernels]
    B --> C[Methodology: Scale matching, tightness proofs & Rebolledo's martingale theorem]
    C --> D[Computational & Theoretical Derivation: Uniqueness for affine stochastic Volterra equations]
    D --> E[Key Finding 1: Weak convergence to coupled stochastic Volterra system driven by independent Brownian motions]
    D --> F[Key Finding 2: Autonomous rough fractional diffusion driving 2nd component via Mittag-Leffler convolution kernel]
    D --> G[Key Finding 3: Short-time decorrelation governed by rougher component & time-rescaled kernels without scale matching]