Non-Parametric Estimation of Multi-dimensional Marked Hawkes Processes
Sobin Joseph Shashi Jain
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.