Non-Parametric Estimation of Multi-dimensional Marked Hawkes Processes

TL;DR

This work addresses the non-parametric estimation of multi-dimensional marked Hawkes processes by learning a joint time–mark kernel, a regime previously approached primarily in a parametric or decoupled fashion. It proposes two neural architectures—SNH with marks for linear kernels and NNNH with marks for non-linear kernels—and uses a Gaussian Mixture Model to model mark distributions, optimizing the log-likelihood with stochastic gradient methods to enable online updates. The approach is validated on synthetic data with known ground truth and applied to high-frequency cryptocurrency order-book data, revealing time and volume dependencies in market microstructure. The results demonstrate accurate kernel recovery, improved predictive capability over baselines, and practical applicability to real-world, high-frequency data analysis. This work advances interpretability and adaptability of marked Hawkes models in finance and other domains with marked events.

Abstract

An extension of the Hawkes process, the Marked Hawkes process distinguishes itself by featuring variable jump size across each event, in contrast to the constant jump size observed in a Hawkes process without marks. While extensive literature has been dedicated to the non-parametric estimation of both the linear and non-linear Hawkes process, there remains a significant gap in the literature regarding the marked Hawkes process. In response to this, we propose a methodology for estimating the conditional intensity of the marked Hawkes process. We introduce two distinct models: \textit{Shallow Neural Hawkes with marks}- for Hawkes processes with excitatory kernels and \textit{Neural Network for Non-Linear Hawkes with Marks}- for non-linear Hawkes processes. Both these approaches take the past arrival times and their corresponding marks as the input to obtain the arrival intensity. This approach is entirely non-parametric, preserving the interpretability associated with the marked Hawkes process. To validate the efficacy of our method, we subject the method to synthetic datasets with known ground truth. Additionally, we apply our method to model cryptocurrency order book data, demonstrating its applicability to real-world scenarios.

Figures (11)

• Figure 1: Comparison between Hawkes process and Marked Hawkes process
• Figure 2: Flowchart showing process for estimation of Marked Hawkes process
• Figure 3: (a) describes the theoretical and estimated kernel obtained using SNH with marks for a one-dimensional Linear Hawkes process, while (b) presents the absolute error between the theoretical and estimated kernel expressed as $|\hat{\phi} - \phi|$
• Figure 4: QQ plot providing a comparative analysis between SNH and SNH with marks estimation
• Figure 5: (a) shows the theoretical kernel and kernel estimated using NNNH with Hawkes for a one-dimensional Non-linear Hawkes process and (b) depicts the absolute error between the theoretical and estimated kernel

Theorems & Definitions (1)

• Remark 3.1