What are the closed-form expressions for the moments of the (exponential) Hawkes process?

Question

I understand that in the method of moments, one takes theoretical expressions for the first several moments in one vector, and empirical moments in another vector, and then finds the parameters that minimizes the difference between them via iteration. What are the expressions for the moments of the Hawkes process with the standard mixture of exponential kernels? I found one paper with expressions for the cumulants, and that the moments can be expressed in terms of the cumulants but it's still not that clear to me.https://arxiv.org/abs/1409.5353 The kernel I'm referring to is

$\lambda (t) = \mu + \sum_{t_i < t} \sum_{j = 1}^P \alpha_j e^{- \beta_j (t - t_i)}$

The $n$-th order cumulant density is defined by $ k^{(i_1, \ldots, i_n)} = \sum_{T \in \mathcal{T}_{(i_1, \ldots, i_n)}^m} w (T) $

where $w (T)$ is the weight of the tree T, defined as the product of the weight of all its edges, times the weight of the root $m$, defined as being equal to $\lambda^m$.

Answer 1

Da Fonseca and Zaatour (2014) consider the case $P=1$ and provide closed-form expressions for the first four moments if $t\to\infty$.

1st moment as example (p. 553, Equation (20)):

$\lim\limits_{t\to\infty}\mathbb{E}\left[N_{t+\tau}-N_t\right]=\frac{\beta\lambda_\infty}{\beta-\alpha}\tau$

For the 2nd moment please refer to p. 553 Equation (21), for the 3rd and 4th moment to p. 577 and p. 578.

Answer 2

You should be clear on what kind of moments/cumulants you want to estimate. The paper you mentioned focuses on integrated cumulants. For example, it provides the formula of the integral of the covariance function. Those integrated moments only depend on the mean number of events and integrals of the kernel functions (this is the strength of the paper). Estimators of the first integrated cumulants are provided on Equations (11), (12) and (13) of the article Uncovering Causality from Multivariate Hawkes Integrated Cumulants (I'm one of the authors).

You should notice that estimators of moments you're looking for do not depend on your model, but only on your data. The expected counterparts of these estimators depend on the model, and matching the two forms help you find the parameters of your model. However, on the references Hawkes Process: Fast Calibration, Application to Trade Clustering and Diffusive Limit (see Chapter 2) and Modeling Financial Contagion Using Mutually Exciting Jump Processes (see Appendix), the formulas of the expected moments only concern the single exponential parametrization.

If your final goal is just to estimate the parameters of a multivariate Hawkes process with exponential kernels, you should have a look to this brand new library (developed by smart guys of my team): https://github.com/X-DataInitiative/tick. Among others, it provides a bunch of algorithms that fit Hawkes process in either the parametric or nonparametric case, even with the sum of exponential parametrization.