Hawkes processes

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

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

Tools

• Fit Hawkes and HawkesN Processes with AMPL and Ipopt • evently

References