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Multivariate Self-Exciting Processes with Dependencies

Caroline Hillairet, Thomas Peyrat, Anthony Réveillac

TL;DR

This work introduces Multivariate Self-Exciting Processes with Dependencies (MSPD) as a unified framework to model cross-dependent frequency and severity in risk settings, using a Poisson imbedding representation with an extra size-dimension. It develops a rigorous toolkit based on Malliavin calculus and Mecke’s formula to derive shifted-process representations and explicit correlation formulas, including a pseudo-chaotic expansion for counting components. The main contributions are the formal MSPD definition, the pseudo-chaotic expansion, and general (as well as separable-kernel) expressions for expectations and covariances, enabling stress-testing and risk-valuation applications in finance and insurance. The approach supports analytical tractability for cross-dependent risk measures and sets the stage for higher-order moment pricing and contract evaluation in future work.

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.

Multivariate Self-Exciting Processes with Dependencies

TL;DR

This work introduces Multivariate Self-Exciting Processes with Dependencies (MSPD) as a unified framework to model cross-dependent frequency and severity in risk settings, using a Poisson imbedding representation with an extra size-dimension. It develops a rigorous toolkit based on Malliavin calculus and Mecke’s formula to derive shifted-process representations and explicit correlation formulas, including a pseudo-chaotic expansion for counting components. The main contributions are the formal MSPD definition, the pseudo-chaotic expansion, and general (as well as separable-kernel) expressions for expectations and covariances, enabling stress-testing and risk-valuation applications in finance and insurance. The approach supports analytical tractability for cross-dependent risk measures and sets the stage for higher-order moment pricing and contract evaluation in future work.

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.

Paper Structure

This paper contains 20 sections, 11 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Multivariate Self-Exciting Processes with Dependencies
  3. Configuration space and the Poisson measure
  4. Kernels and convolutions
  5. Definition of a Multidimensional Self-exciting Process with Dependencies (MSPD)
  6. Elements of Malliavin's calculus
  7. Spaces and the add point operator
  8. The Malliavin derivative
  9. Factorial measures and iterated integrals
  10. The Mecke formula
  11. Pseudo-chaotic expansion for counting processes
  12. Around the pseudo-chaotic expansion of $\boldsymbol{Z}$
  13. Shifted processes
  14. Expectation and correlations
  15. Expectation of the process $\boldsymbol{Z}$ and its shifted version
  16. ...and 5 more sections

Key Result

Proposition 2.8

Consider a vector $\boldsymbol{\mu}:=(\boldsymbol{\mu}^{i})_{1\leq i \leq d}$ where each component $\boldsymbol{\mu}^{i}$ is a function from $\mathbb R^+$ to $\mathbb R^+$ and let $\boldsymbol{\varphi}$ a self-excitation $d$-kernel. Then the system of SDEs admits a unique $\mathbb{F^N}-$predictable solution $\boldsymbol{\lambda}$.

Theorems & Definitions (39)

  • Definition 2.2: d-kernel
  • Definition 2.3: Separable d-kernel
  • Definition 2.5: Non-explosive $d$-kernel
  • Definition 2.6: Self-excitation d-kernel
  • Definition 2.7: See e.g. Bacryetal2013
  • Proposition 2.8: Intensity process
  • proof
  • Definition 2.9: Multidimensional Self-exciting Process with Dependencies : MSPD
  • Remark 2.10
  • Remark 2.11: $\boldsymbol{\lambda}$ as an MSPD
  • ...and 29 more