Hawkes processes

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

$$\mu (t)=\mu +\sum _{1\le j\le p}{\omega }_i{\psi }_{{\nu }_j}(t-t_j)$$

for some set of kernel bandwidths $\{{\nu }_j{\}}_{1\le j\le p}$, kernel weights $\{{\omega }_{{\nu }_j}{\}}_{1\le j\le p}$, kernel locations $\{{\tau }_j{\}}_{1\le j\le p}$.

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

$${\psi }_{\nu }(t):=\frac{{\mathbb{I}}_{0<t\le \nu }}{\nu }$$

giving the following background intensity

$$\mu (t)=\mu +\sum _{1\le j\le p}{\omega }_j\frac{{\mathbb{I}}_{(0,{\nu }_j]}(t-{\tau }_j)}{{\nu }_j}.$$

I augment the parameter vector to include the kernel weights ${\theta }^{\prime }:=(\mu ,\eta ,\kappa ,\mathit{\omega }).$ We could also try to infer kernel locations and bandwidths.

The hypothesized generating model now has conditional intensity process

$${\lambda }_{{\theta }^{\prime }}(t|{\mathcal{F}}_t)=\mu +\sum _{j=2}^n{\omega }_j{\mathbb{I}}_{[{\tau }_{j-1},{\tau }_j)}(t)+\eta \sum _{t_i<t}{\varphi }_{\kappa }(t-t_i).$$

1 Tools

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

2 References