Paper: arXiv 2503.15958

Authors: Caroline Hillairet, Thomas Peyrat, Anthony Réveillac

Abstract

This paper introduces the class of multidimensional self-exciting processes with dependencies (MSPD), which is a unifying writing for a large class of processes: counting, loss, intensity, and also shifted processes. The framework takes into account dynamic dependencies between the frequency and the severity components of the risk, and therefore induces theoretical challenges in the computations of risk valuations. We present a general method for calculating different quantities related to these MSPDs, which combines the Poisson imbedding, the pseudo-chaotic expansion and Malliavin calculus. The methodology is illustrated for the computation of explicit general correlation formula.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a highly sophisticated mathematical extension of multivariate Hawkes processes using Malliavin calculus, pseudo-chaotic expansions, and Poisson imbedding. However, it is purely theoretical with no empirical data analysis, backtesting, or code implementation provided in the excerpt.

Research Flowchart

  flowchart TD
    A[Research Goal: Compute risk valuations for processes with frequency-severity dependencies] --> B[Input: Multidimensional Self-Exciting Processes with Dependencies]
    B --> C[Methodology: Poisson Imbedding]
    C --> D[Methodology: Pseudo-Chaotic Expansion & Malliavin Calculus]
    D --> E[Computational Process: Derivation of Explicit Valuations]
    E --> F[Outcomes: Unified MSPD Framework & Explicit Correlation Formulas]

Cited References (15)

  1. Normal approximation of Functionals of Point Processes: Application to Hawkes Processes. (2024) — link
  2. Explicit correlations for the Hawkes processes (2023) — link
  3. On the chaotic expansion for counting processes (2022) — link
  4. An expansion formula for Hawkes processes and application to cyber-insurance derivatives (2021) — link
  5. Multivariate Hawkes process for cyber insurance (2020) — link
  6. Shot-Noise Processes in Finance (2016) — link
  7. Limit theorems for nearly unstable Hawkes processes (2013) — link
  8. Some limit theorems for Hawkes processes and application to financial statistics (2013) — link
and 7 more
  1. Contagion in cyber security attacks (2012) — link
  2. Multivariate Hawkes processes: an application to financial data (2011) — link
  3. Functional Ito calculus and stochastic integral representation of martingales (2010) — link
  4. Affine Point Processes and Portfolio Credit Risk (2010) — link
  5. Stochastic Analysis in Discrete and Continuous Settings: With Normal Martingales (2009) — link
  6. Stochastic Analysis for Poisson Point Processes (2016) — link
  7. Sur l’approximation des processus de Hawkes : des séries temporelles aux théorèmes centraux limites quantitatifs — link

0 of 15 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.