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

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.