The functional discrete-time approximation of marked Hawkes risk processes

The marked Hawkes risk process is a compound point process where the occurrence and amplitude of past events impact the future. Since data in real life are acquired over a discrete time grid, we propose a strong discrete-time approximation of the continuous-time risk process obtained by embedding from the same Poisson measure. We then prove trajectorial convergence results in both fractional Sobolev spaces and the Skorokhod space, hence extending the theorems proven in Huang and Khabou ((2023). Stoch. Process. Appl. 161 , 201–241) and Kirchner ((2016). Stoch. Process. Appl. 126 (8), 2494–2525). We also provide upper bounds on the convergence speed with explicit dependence on the size of the discretization step, the time horizon, and the regularity of the kernel.