Exponential moments for Hawkes processes under minimal assumptions
In this article the author proves that a stationary Hawkes process has exponential moments under the minimal spectral radius condition given by \textit{P. Brémaud} and \textit{L. Massoulié} [Ann. Probab. 24, No. 3, 1563--1588 (1996; Zbl 0870.60043)],\N\[\N\operatorname{SpR} (\|h^{m}_{m^{\prime}}\|_{1})_{m,m^{\prime} \in M}<1 \N\]\Nwhere \(h^{m}_{m^{\prime}}\) is the Lebesgue integrable interaction function. Under the spectral radius condition, a linear stationary Hawkes process \(N = (N^{1},N^{2},\dots,N^{M})\) with intensity\N\[\N\lambda^{m}_{t} = \mu_{m} + \sum_{m^{\prime} \in M} \int_{-\infty}^{t-} h_{m^{\prime}}^{m}(t-s)dN_{s}^{m^{\prime}}\N\]\Nexists and has exponential moments. The author also considers a cluster representation such as the Galton-Watson tree and gives a functional version bound.
scientific article; zbMATH DE number 7960010
scientific article