Anatoliy Swishchuk, University of Calgary

Title: Hawkes Processes and their Applications in Finance and Insurance.

Abstract: Hawkes processes are class of stochastic counting processes that have been applied in diverse areas, from earthquake modelling to financial analysis. They are point processes whose defining characteristic is that they ‘self-excite’, meaning that each arrival increases the rate of future arrivals for some period of time. Thus, the Hawkes process is a mathematical model for these ‘self-exciting’ processes, named after its creator Alan G. Hawkes (1971), and it is a non-Markovian extension of the Poisson process. Hawkes models are becoming more and more popular in the domains of high frequency finance and insurance. The talk is devoted to the Hawkes processes, their applications in finance and insurance, and consists of three parts. The first part introduces the Hawkes processes and describes their properties. The second part is devoted to the modelling of trading activities in high frequency finance, namely, modelling of limit order books, and to the justification of our modelling by presenting some numerical examples using real data. The third part focuses on application of Hawkes processes in insurance: we study a risk model with claim arrivals based on so called general compound Hawkes process, and show that it is suitable to model empirical real insurance data set. The investment problem of an insurer in an incomplete market whose claims arrive according to the Hawkes process will addressed as well. (The second part of the talk is based on our joint paper with B. Remillad and R. Elliott, and the third part-on our joint paper with R. Zagst and G. Zeller (TUM, Munich, Germany).)