Scaling Limits of Bivariate Nearly-Unstable Hawkes Processes and Applications to Rough Volatility

TL;DR

This paper establishes a limit theorem for nearly unstable bivariate Hawkes processes with heterogeneous roughness, revealing a coupled stochastic Volterra system with distinct fractional diffusions.

Contribution

It introduces the first multivariate nearly unstable Hawkes process model with different roughness exponents and characterizes its limit as a coupled stochastic Volterra system.

Findings

The limit process consists of a rough fractional diffusion and a coupled component driven by its own and the first component's noise.

The correlation between components vanishes at a polynomial rate depending on the roughness.

Without scale-matching, the limit involves a time-rescaled convolution kernel instead of the original cross-kernel.

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.