Self-exciting jump processes and their asymptotic behaviour

Spectra of some self - exciting and mutually exciting point processes

In recent years methods of data analysis for point processes have received some attention, for example, by Cox & Lewis (1966) and Lewis (1964). In particular Bartlett (1963a, b) has introduced methods of analysis based on the point spectrum. Theoretical models are relatively sparse. In this paper the theoretical properties of a class of processes with particular reference to the point spectrum ...

Self-exciting point processes with applications in finance and medicine

Asymptotic behaviour of gossip processes and small world networks

Both small world models of random networks with occasional long range connections and gossip processes with occasional long range transmission of information have similar characteristic behaviour. The long range elements appreciably reduce the effective distances, measured in space or in time, between pairs of typical points. In this paper, we show that their common behaviour can be interpreted...

Symmetric Jump Processes and their Heat Kernel Estimates

We survey the recent development of the DeGiorgi-Nash-Moser-Aronson type theory for a class of symmetric jump processes (or equivalently, a class of symmetric integro-differential operators). We focus on the sharp two-sided estimates for the transition density functions (or heat kernels) of the processes, a priori Hölder estimate and parabolic Harnack inequalities for their parabolic functions....

Jump processes

Although historically models in mathematical finance were based on Brownian motion and thus are models with continuous price paths, jump processes play now a key role across all areas of finance (see e.g. [5]). One reason for this move into a new class of processes is that because of their distributional properties diffusions in many cases cannot provide a realistic picture of empirically obser...