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Malliavin calculcus for a Hawkes process

Alexandre Popier, Laurent Denis, Dorian Cacitti-Holland

TL;DR

This work develops a local Malliavin calculus on nonlinear Hawkes processes by perturbing jump times and constructing a local Dirichlet form with a gradient and divergence. Using a Cameron–Martin-type time change, it establishes absolute continuity criteria for Hawkes-driven functionals and derives integration-by-parts formulas, enabling density results for Hawkes SDEs and Malliavin-based Greeks. The framework yields a precise, multi-layer Dirichlet structure (through directional derivatives and a full Dirichlet form via a Hilbert basis) and provides local and global density criteria under explicit conditions. Together, these results equip stochastic modeling with regularity tools for Hawkes-driven dynamics and provide practical pathways for sensitivity analysis in finance.

Abstract

We develop a Malliavin calculus for nonlinear Hawkes processes in the sense of Carlen and Pardoux. This approach, based on perturbations of the jump times of the process, enables the construction of a local Dirichlet form. As an application, we establish criteria for the absolute continuity of solutions to stochastic differential equations driven by Hawkes processes. We also derive sensitivity formulas for the valuation of financial derivatives with respect to model parameters.

Malliavin calculcus for a Hawkes process

TL;DR

This work develops a local Malliavin calculus on nonlinear Hawkes processes by perturbing jump times and constructing a local Dirichlet form with a gradient and divergence. Using a Cameron–Martin-type time change, it establishes absolute continuity criteria for Hawkes-driven functionals and derives integration-by-parts formulas, enabling density results for Hawkes SDEs and Malliavin-based Greeks. The framework yields a precise, multi-layer Dirichlet structure (through directional derivatives and a full Dirichlet form via a Hilbert basis) and provides local and global density criteria under explicit conditions. Together, these results equip stochastic modeling with regularity tools for Hawkes-driven dynamics and provide practical pathways for sensitivity analysis in finance.

Abstract

We develop a Malliavin calculus for nonlinear Hawkes processes in the sense of Carlen and Pardoux. This approach, based on perturbations of the jump times of the process, enables the construction of a local Dirichlet form. As an application, we establish criteria for the absolute continuity of solutions to stochastic differential equations driven by Hawkes processes. We also derive sensitivity formulas for the valuation of financial derivatives with respect to model parameters.
Paper Structure (19 sections, 30 theorems, 257 equations)

This paper contains 19 sections, 30 theorems, 257 equations.

Table of Contents

  1. Introduction
  2. Breakdown.
  3. Framework and directional derivation
  4. Setting and density of the jump times
  5. Cameron-Martin space and directional derivation
  6. Absolute continuity of $\mathbb{P}^\varepsilon$ w.r.t. $\mathbb{P}$
  7. Integration by parts in Bismut's way
  8. Directional Dirichlet space
  9. The local Dirichlet form
  10. Definition using a Hilbert basis
  11. Divergence operator by duality
  12. Absolute continuity criterion
  13. Local criterion
  14. Global criterion
  15. Applications
  16. ...and 4 more sections

Key Result

Lemma 2.2

The distribution of $(T_1, \cdots, T_n)$ conditionally to $\{N_T = n\}$ has a density with, for any $t = (t_1, \cdots, t_n) \in \mathbb{R}^n$,

Theorems & Definitions (76)

  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Proposition 2.8
  • ...and 66 more