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Some limit theorems for locally stationary Hawkes processes

Thomas Deschatre, Pierre Gruet, Antoine Lotz

TL;DR

The paper develops a functional law of large numbers and a functional central limit theorem for a class of locally stationary multivariate Hawkes processes with time-dependent reproduction rates, addressing non-convolution Volterra kernels by integrating classical martingale techniques with grid-refinement methods. It provides explicit LLN and FTCL limit forms in terms of $\mathbf{K}(x)=g(x)\int_0^{\infty}\varphi(s)\mathrm{d}s$ and $\boldsymbol{\Sigma}(x)$, and demonstrates applications to financial statistics, including closed-form price distortions under liquidity constraints. The work extends existing asymptotic results to non-stationary, non-convolution settings and offers practical insights for inference and modeling in finance, with potential extensions to covariates and marked processes. Overall, it lays a rigorous theoretical foundation for analyzing and applying locally stationary Hawkes processes where the reproduction mechanism evolves deterministically in time.

Abstract

We prove a law of large numbers and functional central limit theorem for a class of multivariate Hawkes processes with time-dependent reproduction rate. We address the difficulties induced by the use of non-convolutive Volterra processes by recombining classical martingale methods introduced in Bacry et al. [3] with novel ideas proposed by Kwan et al. [19]. The asymptotic theory we obtain yields useful applications in financial statistics. As an illustration, we derive closed-form expressions for price distortions under liquidity constraints.

Some limit theorems for locally stationary Hawkes processes

TL;DR

The paper develops a functional law of large numbers and a functional central limit theorem for a class of locally stationary multivariate Hawkes processes with time-dependent reproduction rates, addressing non-convolution Volterra kernels by integrating classical martingale techniques with grid-refinement methods. It provides explicit LLN and FTCL limit forms in terms of and , and demonstrates applications to financial statistics, including closed-form price distortions under liquidity constraints. The work extends existing asymptotic results to non-stationary, non-convolution settings and offers practical insights for inference and modeling in finance, with potential extensions to covariates and marked processes. Overall, it lays a rigorous theoretical foundation for analyzing and applying locally stationary Hawkes processes where the reproduction mechanism evolves deterministically in time.

Abstract

We prove a law of large numbers and functional central limit theorem for a class of multivariate Hawkes processes with time-dependent reproduction rate. We address the difficulties induced by the use of non-convolutive Volterra processes by recombining classical martingale methods introduced in Bacry et al. [3] with novel ideas proposed by Kwan et al. [19]. The asymptotic theory we obtain yields useful applications in financial statistics. As an illustration, we derive closed-form expressions for price distortions under liquidity constraints.

Paper Structure

This paper contains 20 sections, 27 theorems, 141 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation and relation to other works
  3. Main contribution
  4. Locally stationary Hawkes processes
  5. Definition.
  6. Stability of the process.
  7. Law of large number and central limit theorem.
  8. Application to financial statistics
  9. A model for prices under liquidity constraints
  10. Numerical experiments
  11. Discussion
  12. Preparation for the proofs
  13. On Volterra equations
  14. Proof of Proposition \ref{['prop:deterministic_convergence']}
  15. A trend plus noise decomposition of the intensity
  16. ...and 5 more sections

Key Result

Proposition 1

For any $(\mathcal{F}_t^\pi)$--progressively measurable process $(\lambda_ {k,t})$, the process defined by admits $(\lambda_{k,t})$ as its intensity.

Figures (2)

  • Figure 1: Theoretical limit of the normalized process $T^{-1}N^T_t$ as a function of the normalized time $\frac{t}{T}$ (solid line), as compared to its empirical estimation $n^{-1} \sum_{k=1}^n T^{-1} N^{k,T}_t$ on $n=500$ independent simulations $(N^{k,T}_t)$ of the process with $T=200$ (dotted line), with a Gaussian reproduction function (left) and a linear reproduction function (right).
  • Figure 2: Reproduction function $\alpha$ (left), Expected return (middle), and Slippage (Right) for a locally stationary Hawkes prices model. Expected returns are computed from Theorem \ref{['Th:LLN']} and compared to the Monte-Carlo estimator $n^{-1} \sum_{k=1}^n T^{-1} P^{k,T}_t$ for $n=3000$ independent simulations.

Theorems & Definitions (52)

  • Proposition 1
  • Definition 1
  • Remark 1
  • Lemma 1
  • Theorem 1: Law of Large Numbers
  • Theorem 2: Central limit theorem
  • Corollary 1
  • Remark 2
  • Remark 3
  • Proposition 2
  • ...and 42 more