Scaling Limit Theorems for Multivariate Hawkes Processes and Stochastic Volterra Equations with Measure Kernel

TL;DR

This paper establishes scaling limit theorems for multivariate Hawkes processes, showing their convergence to solutions of stochastic Volterra equations, and provides explicit representations of their Fourier-Laplace functionals.

Contribution

It introduces a new approach to prove convergence of multivariate Hawkes processes to stochastic Volterra equations under mild conditions, with explicit functional representations.

Findings

Multivariate Hawkes processes converge to stochastic Volterra solutions after scaling.

Explicit exponential-affine representations of Fourier-Laplace functionals are derived.

Regularity and alternative representations of limit processes are analyzed.

Abstract

This paper is devoted to establishing the full scaling limit theorems for multivariate Hawkes processes. Under some mild conditions on the exciting kernels, we develop a new way to prove that after a suitable time-spatial scaling, the asymptotically critical multivariate Hawkes processes converge weakly to the unique solution of a multidimensional stochastic Volterra equation with convolution kernel being the potential measure associated to a matrix-valued extended Bernstein function. Also, based on the observation of their affine property and generalized branching property, we provide an exponential-affine representation of the Fourier-Laplace functional of scaling limits in terms of the unique solutions of multidimensional Riccati-Volterra equations with measure kernel. The regularity of limit processes and their alternate representations are also investigated by using the potential theory of L\'evy subordinators.