Perfect simulation of Hawkes processes

Abstract

Our objective is to construct a perfect simulation algorithm for unmarked and marked Hawkes processes. The usual straightforward simulation algorithm suffers from edge effects, whereas our perfect simulation algorithm does not. By viewing Hawkes processes as Poisson cluster processes and using their branching and conditional independence structures, useful approximations of the distribution function for the length of a cluster are derived. This is used to construct upper and lower processes for the perfect simulation algorithm. A tail-lightness condition turns out to be of importance for the applicability of the perfect simulation algorithm. Examples of applications and empirical results are presented.

Modeling Neuronal Firing in Epilepsy: Fitting Hawkes Processes to Single-Unit Activity

“…Taking expectation on both side, and setting λ = Eλ t , we get the equation λ = µ + cλ , and the necessary (and sufficient, see [11]) condition for the existence of a Hawkes process satisfying (2):…”

Modelling Trades-Through in a Limit Order Book Using Hawkes Processes

“…Note also that simulation algorithms of Hawkes processes are easily available (see e.g. Ogata, 1981;Moller and Rasmussen, 2005, and Appendix A), allowing further developments for the use of the estimated models. These practical properties make Hawkes processes good candidates for the study of the clustering of events empirically observed in order book models (Hautsch, 2004;Bowsher, 2007;Large, 2007;Bacry et al, 2012).…”

“…Taking a different point of view, a more recent work by Moller and Rasmussen (2005) claims to provide better quality simulations, without edge effects. In this appendix however, we describe the simple thinning algorithm in a multidimensional setting, because of its intuitive progression and its ability to be straightforwardly generalized to any point process described with its conditional intensity.…”

Performance of Information Criteria for Selection of Hawkes Process Models of Financial Data

“…As there is an intensity-based as well as cluster-based definition, there exist various simulation and estimation techniques which take advantage of either perspective on Hawkes processes. To mention a few, for simulation we have the thinning approach (Ogata 1981), the timechange approach based on the random time-change theorem (Meyer 1971) and applied specifically to Hawkes processes for instance in (Ozaki 1979), exact simulation (Dassios and Zhao 2013) and perfect simulation (Møller and Rasmussen 2005). Concerning estimation techniques, the standard maximum likelihood approach can for example be found in Ozaki (1979).…”