Affine pure-jump Volterra fields

Abstract

We study a class of non-negative spatio-temporal random fields that exhibit self-exciting clustering, which we refer to as affine pure-jump Volterra fields. They are defined via stochastic integration of a Volterra kernel against a thinned Poisson random measure and the thinning is such that the field admits a representation as a stochastic integral of the same kernel, but against a random measure whose compensator has a density that is pointwise affine in the field itself. This representation leads to affine transform formulas characterizing the Laplace transform of functionals of the field up to the solution of a deterministic non-linear Volterra integral equation. Affine pure-jump Volterra random fields extend a non-negative finite-first-moment subclass of pure-jump affine Volterra processes to the random-field setting. The framework includes non-negative finite-variation ambit fields, marked Hawkes systems, and branching-type models as special cases, and we make these connections explicit through examples.