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Abstract

We consider multivariate Hawkes processes with baseline conditional intensities and reproduction functions that depend on time. In this case, the model is characterized via the conditional intensity function which we want to estimate. Thus, a class of locally stationary processes is defined. The discussed estimation procedure of the vector of time-dependent baseline intensities and vector of reproduction functions is grounded on a localized criterion. Theory on stationary Hawkes processes is extended to develop asymptotic theory for the estimator in the locally stationary model. Simulation studies round off the considerations. Furthermore, we consider the option for local alignment of locally stationary Hawkes processes. The previous work enables the possibility to formulate testing results. We observe two Hawkes processes and test whether they are realizations of the same underlying immigration and reproduction functions at a fixed point in time. From an alternative point of view, but identically contentwise, one could compare the behaviour of one Hawkes process at two distinct time points. This enables reasearch sorrounding seasonality.