Develop a dependent marked version where mark sizes influence intensities

Develop and analyze a dependent marked version of the proposed multivariate self- and mutually exciting point process with flexible residuals, in which the mark size of each event affects the subsequent λ processes, rather than assuming independent marks.

Background

In the empirical applications, the paper treats the mark sizes (price jump magnitudes) as independent and not influencing future intensities to keep the model tractable. This simplifies estimation and simulation but omits potentially important dependence between mark sizes and subsequent event intensities.

The authors explicitly state that allowing marks to affect the intensity is feasible but increases complexity, and they leave this dependent marked extension for future research.

References

Although it is feasible to implement a dependent marked model in which the mark size affects the λ process in the proposed model, such an approach would increase the model complexity and is left for future research.

— Self and mutually exciting point process embedding flexible residuals and intensity with discretely Markovian dynamics  (2401.13890 - Lee, 2024) in Subsubsection 'Multivariate model and simulation', Section 3.2 'Sparsely observed data'

Because the proof uses the independent Poisson structure of the mapped weights, however, its extension to self-exciting histories is established here numerically rather than theoretically. A formal derivation would need to control the coupled evolution of the thresholded counts and weights as q varies, rather than only their fixed-threshold marginal transforms in equations~eq:cluster_functional--eq:stationary_functional.

— Finite-Horizon Triggering Cores of Earthquake Sequences  (2609.37213 - Sornette et al., 29 Sep 2026) in Section 7, Discussion, subsection “New information provided by the finite-horizon triggering core size”