ANR HAPPY : HAwkes Point Processes: better understanding the effects of nonlineariY
ANR JCJC (ANR-23-CE40-0007, 2024-2028)
Hawkes processes are currently used in a growing number of fields such as neuroscience, finance and insurance.
These point processes model the successive occurrences of events and their influence on the probability of future occurrences.
The diversity of applications requires to consider general models including nonlinearity in the time dependence, inhibition between events or an interaction network.
New mathematical questions emerge from these nonlinear Hawkes processes, one of the major difficulties coming from the complex dependence structures between events generated by the nonlinearity or by the inhibition.
Recent advances have been made regarding the long-time properties of a class of nonlinear Hawkes procesess with inhibition, by highlighting some renewal property for these processes.
These results open a promising new direction to study Hawkes processes with general inhibition patterns, but the development of statistical applications of these results requires a better understanding of this structure.
Similar questions also arise in a multivariate setting, as inhibition plays an important role in the regulation of activity in complex networks.
Moreover, if many techniques have been developed to infer the parameters of Hawkes processes, few tools are available to test for the presence of regulation or of temporal inhibition in networks and to identify the underlying structure of the observed networks.
A further difficulty and a longstanding issue is to understand the interplay between inhibition and the size and geometry of the network, in particular in the context of mean-field interactions. %
