Functional Laws of Large Numbers for Marked Hawkes Processes and Compound Marked Hawkes Processes
Tomasz R. Bielecki, Jacek Jakubowski, Mariusz iewȩgłowski, Anatoliy Swishchuk
TL;DR
This work extends functional laws of large numbers to marked Hawkes processes with general (not necessarily discrete) marks and to associated marked compound Hawkes processes. By leveraging a thinning representation and Volterra-type equations for the intensity, the authors derive $L^2$-convergent LLNs for the primary counting process $N$ and for two compounded variants $C^A$ and $D^\varphi$, with explicit deterministic limits that depend on the base intensity $m$, the stationary resolvent $R$, and the mark distribution $Q$. The results unify and generalize prior discrete-mark analyses (BDHM) and continuous-mark settings (HORST202194), and they illustrate an insurance ruin application showing how the net-profit condition adapts to the self-exciting structure. The findings provide a rigorous asymptotic framework for long-run averages of event counts and aggregated mark-dependent quantities, enabling robust risk assessment in domains like cyber-insurance. The paper also outlines concrete paths for future work, including uniform convergence, almost sure results, and functional central limit theorems for these marked processes.
Abstract
We give functional laws of large numbers for a class of marked Hawkes processes and marked compound Hawkes processes with a general mark space. Our results provide some complement to those presented previously in the literature. As an example we provide an application to analysis of time limit of an insurance ruin process.