# Hawkes processes

December 22, 2019 — December 22, 2019

At the intersection of point processes and branching processes can be found the Hawkes process.

The classic is the univariate, temporal, linear Hawkes process.git Recall the log likelihood of a generic point process, with occurrence times \(\{t_i\}.\)

\[ \begin{aligned} L_\theta(t_{1:N}) &:= -\int_0^T\lambda^*_\theta(t)dt + \int_0^T\log \lambda^*_\theta(t) dN_t\\ &= -\int_0^T\lambda^*_\theta(t)dt + \sum_{j} \log \lambda^*_\theta(t_j) \end{aligned} \]

\(\lambda^*(t)\) is shorthand for \(\lambda^*(t|\mathcal{F}_t)\), and we call this the intensity. This term is what distinguishes various point processes. For the Hawkes process is in particular we have

\[ \lambda^*(t) = \mu + \int_{-\infty}^t \eta\phi(t-s)dNs. \] where \(\phi_\kappa(t)\) is the influence kernel with parameter \(\kappa\), \(\eta\) the branching ratio, and

### 0.1 Time-inhomogeneous extension

Partial notes to an extension that I have looked at. Introduce an additional convolution kernel \(\psi\), and functions of the form

\[ \mu(t) = \mu + \sum_{1 \leq j \leq p}\omega_i\psi_{\nu_j}(t-t_j) \]

for some set of kernel bandwidths \(\{\nu_j\}_{1 \leq j \leq p}\), kernel weights \(\{\omega_{\nu_j}\}_{1 \leq j \leq p}\), kernel locations \(\{\tau_j\}_{1 \leq j \leq p}\).

There are many kernels available. We start with the *top hat* kernel, the piecewise-constant function.

\[ \psi_{\nu}(t):= \frac{\mathbb{I}_{0< t \leq \nu}}{\nu} \]

giving the following background intensity

\[ \mu(t) = \mu + \sum_{1\leq j\le p}\omega_j\frac{\mathbb{I}_{(0, \nu_j]}(t-\tau_j)}{\nu_j}. \]

I augment the parameter vector to include the kernel weights \(\theta':=( \mu,\eta,\kappa, \boldsymbol\omega).\) We could also try to infer kernel locations and bandwidths.

The hypothesized generating model now has conditional intensity process

\[ \lambda_{\theta'}(t|\mathcal{F}_t) = \mu + \sum_{j=2}^n \omega_j \mathbb{I}_{[\tau_{j-1},\tau_j)}(t) + \eta \sum_{t_i< t}\phi_\kappa(t-t_i). \]

## 1 Tools

## 2 References

*PMLR*.

*Advances in Imaging and Electron Physics*.

*The European Physical Journal B*.

*Quantitative Finance*.

*Stochastic Processes and Their Applications*, A Special Issue on the Occasion of the 2013 International Year of Statistics,.

*arXiv:1412.7096 [q-Fin, Stat]*.

*Quantitative Finance*.

*IEEE Transactions on Information Theory*.

*Journal of Econometrics*.

*Journal of Applied Probability*.

*Advances in Applied Probability*.

*Advances in Applied Probability*.

*Journal of Applied Probability*.

*Journal of Computational and Graphical Statistics*.

*Electronic Communications in Probability*.

*Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining*. KDD ’15.

*Journal of Time Series Analysis*.

*Journal of Applied Probability*.

*Journal of International Money and Finance*, Understanding International Commodity Price Fluctuations,.

*Physical Review E*.

*Communications in Nonlinear Science and Numerical Simulation*.

*Advances in Imaging and Electron Physics*.

*Psychometrika*.

*Psychometrika*.

*Bernoulli*.

*The European Physical Journal B*.

*Physical Review E*.

*Journal of the Royal Statistical Society. Series B (Methodological)*.

*Biometrika*.

*Journal of Applied Probability*.

*The Annals of Applied Probability*.

*Physical Review E*.

*arXiv:1211.4039 [Math]*.

*Proceedings of the 13th International Conference on Web Search and Data Mining*. WSDM ’20.

*The Annals of Applied Probability*.

*arXiv:1507.02822 [Math, q-Fin, Stat]*.

*Preprint*.

*Twenty-Eighth AAAI Conference on Artificial Intelligence*.

*Proceedings of the 25th ACM International Conference on Information and Knowledge Management*. CIKM ’16.

*Journal of Physics A: Mathematical and Theoretical*.

*The Annals of Applied Statistics*.

*Advances in Applied Probability*.

*Methodology and Computing in Applied Probability*.

*Statistics & Probability Letters*.

*Annals of the Institute of Statistical Mathematics*.

*Proceedings of the 2014 Tenth International Conference on Signal-Image Technology and Internet-Based Systems*. SITIS ’14.

*Physical Review E*.

*Methodology and Computing in Applied Probability*.

*Bulletin of the Belgian Mathematical Society - Simon Stevin*.

*The Annals of Statistics*.

*Proceedings of the 2018 World Wide Web Conference*.

*World Wide Web 2017, International Conference on*. WWW ’17.

*arXiv:1101.1611 [Cond-Mat, Physics:physics]*.

*arXiv:1101.5564 [Cond-Mat, Physics:physics]*.

*Proceedings of the 33rd International Conference on International Conference on Machine Learning - Volume 48*. ICML’16.

*Comput. Stat. Data Anal.*

*Proceedings of the 30th International Conference on Machine Learning (ICML-13)*.

*Statistics & Probability Letters*.