This chapter provides the mathematical foundations for stochastic intensity, on which most of the methodology is based. The intensity of an event such as default is its conditional mean arrival rate, measured in events per year, given all information currently available to the observer. The chapter also presents the multi-firm version of the doubly-stochastic hypothesis, under which the sole source of default correlation between two firms is the dependence of their default intensities on common or correlated observable risk factors. The doubly-stochastic property rules out contagion as well as correlation induced by unobservable risk factors. The chapter includes a mathematical device for testing a model of the default intensity processes of a large number of borrowers.
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