Dependent Events and Changes of Time∗

Meyer (1971) showed that any point process whose compensator has continuous paths that increase to ∞ 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 ∞ 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. Timechanged 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. ∗This research is supported by grants from the Global Derivatives Research Group at JP Morgan Chase & Co, and from the Natural Sciences and Engineering Council of Canada, for which we are very grateful. We would like to thank Andrew Abrahams, Eymen Errais, Lionel Pradier and Gael Riboulet for stimulating discussions. †School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY 14853-3801, USA, Phone (607) 255 9140, Fax (607) 255 9129, email: [email protected]. ‡School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY 14853-3801, USA, Phone (607) 255 8842, Fax (607) 255 9129, email: [email protected].