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This study considers a new multi-term urn process that has a correlation inthe same term and temporal correlation. The objective is to clarify therelationship between the urn model and the Hawkes process. Correlation in thesame term is represented by the P\'{o}lya urn model and the temporalcorrelation is incorporated by introducing the conditional initial condition.In the double-scaling limit of this urn process, the self-exciting negativebinomial distribution (SE-NBD) process, which is a marked Hawkes process, isobtained. In the standard continuous limit, this process becomes the Hawkesprocess, which has no correlation in the same term. The difference is thevariance of the intensity function in that the phase transition from the steadyto the non-steady state can be observed. The critical point, at which the powerlaw distribution is obtained, is the same for the Hawkes and the urn processes.These two processes are used to analyze empirical data of financial default toestimate the parameters of the model. For the default portfolio, the resultsproduced by the urn process are superior to those obtained with the Hawkesprocess and confirm self-excitation.

From the multi-terms urn model to the self-exciting negative binomial distribution and Hawkes processes