Том 62 ТЕОРИЯ ВЕРОЯТНОСТЕЙ И ЕЕ ПРИМЕНЕНИЯ 2017 2017 г. c EMBRECHTS P., KIRCHNER M.∗ HAWKES GRAPHS1) Вып у с к 1 В статье введены понятия графа и остовного графа Хоукса. <...> DOI: https://doi.org/10.4213/tvp5093 ∗ETH Z¨ urich, Department of Mathematics, RiskLab, Z¨ matthias.kirchner@math.ethz.ch 1)This work was supported by RiskLab Z¨ urich, Switzerland; e-mail: urich and the Swiss Finance Institute. 164 Embrechts P., Kirchner M. 1. <...> This paper discusses the specification and estimation of multivariate Hawkes point process models from large, multitype eventstream datasets such as neural spike-trains, internet search-queries, or limitorder-book data in high-frequency finance. <...> Our approach uses the notion of a Hawkes skeleton and a Hawkes graph2). <...> TheHawkes process was introduced in [9], [10] as a stationary point process on R whose points are assigned to a finite number of types. <...> The (stochastic) intensity of a Hawkes process depends on the past of the process itself: given the occurrence of an event, the intensities — the expected mean number of events per time unit and event type — typically jump upwards and then decay. <...> In the present paper, we formally introduce the Hawkes graph. <...> TheHawkes graph summarizes the branching structure of a multitype Hawkes point process as a directed graph with weighted vertices and edges. <...> The vertex weights are the corresponding immigration intensities; the weight of an edge (i, j) is the expected number of type-j children events that an type-i event generates. <...> The Hawkes skeleton is the Hawkes graph disregarding the weights. <...> The graph terminology is convenient to describe many relevant aspects of multivariate Hawkes processes such as «ancestor and parent sets», «paths», «path weights», «feedback», «cascades», or «connectivity». <...> In particular, the likelihood approach does not provide tests to decide whether excitement from one event type to another exists at all. <...> We apply an estimation procedure developed in [13]. <...> This is why, in combination with the concept of a Hawkes skeleton and graph, we tackle the numerical difficulties by the following three-step algorithm: 1. <...> Given a large multitype event-stream dataset, we first apply a specific testing scheme to decide whether there is any effect from a specific <...>