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The paper discusses multivariate self- and cross-exciting processes. Wedefine a class of multivariate point processes via their correspondingstochastic intensity processes that are driven by stochastic jumps.Essentially, there is a jump in an intensity process whenever the correspondingpoint process records an event. An attribute of our modelling class is that notonly a jump is recorded at each instance, but also its magnitude. This allowslarge jumps to influence the intensity to a larger degree than smaller jumps.We give conditions which guarantee that the process is stable, in the sensethat it does not explode, and provide a detailed discussion on when thesubclass of linear models is stable. Finally, we fit our model to financialtime series data from the S\&P 500 and Nikkei 225 indices respectively. Weconclude that a nonlinear variant from our modelling class fits the data best.This supports the observation that in times of crises (high intensity) jumpstend to arrive in clusters, whereas there are typically longer times betweenjumps when the markets are calmer. We moreover observe more variability in jumpsizes when the intensity is high, than when it is low.

Multivariate self-exciting jump processes with applications to financial data