Hawkes processes – a brief history, some applications, and an open problem

Hawkes processes are a class of point processes on the real line. Typically, the line represents time and the points represent events. This talk provides an overview of the theoretical developments of Hawkes processes since their introduction in the 1970s.  Furthermore, we give numerous examples for applications to financial event streams where Hawkes processes are often found to be a good fit. It is argued that the reason for this flexibility lies in their autoregressive structure. However, in the linear case, the autoregressive structure is limited as it only allows excitement and not inhibition from past events on the jump rate. An exponential link-function would be a natural choice for a non-linear Hawkes process that allows both excitement and inhibition. It is an open problem whether this loglinear case can yield non-trivial stable versions.