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arXiv · 2412.14459

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

Wei Xu
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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 Lévy subordinators.

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Wei Xu. 2024-12-19. Scaling Limit Theorems for Multivariate Hawkes Processes and Stochastic Volterra Equations with Measure Kernel. https://arxiv.org/abs/2412.14459

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