Dependent Events and Changes of Time
by Kay Giesecke of Cornell University, and
July 7, 2005
Abstract: Meyer (1971) showed that any point process whose compensator has continuous paths that increase to 1 can be time-scaled to a standard Poisson process. In this article we consider the converse to this result. We construct a time change with continuous paths increasing to 1 that transforms a standard Poisson process into a general point process with totally inaccessible arrivals and compensator given by the time change. The time change generates path-dependent or self-affecting point processes whose dynamics depend on the information generated by the arrivals of the process as well as other observable information describing the state of the random environment. The classical Hawkes and doubly stochastic processes are special cases. Time-changed Hawkes processes are shown to combine the best features of these classical families is a flexible and tractable way. We conclude by introducing time-change techniques to multi-name credit modeling. We describe the economy-wide default process as a time-changed Poisson process, whose arrivals are temporally clustered due to contagion and depend on the economic environment. Random thinning generates sub-models for individual firms and portfolios of firms.