Random distribution kernels and three types of defaultable contingent payoffs

Abstract

We introduce the random distribution kernel on a product probability space and obtain the representation results connecting the product and base probability spaces. Using the random variable with the random distribution kernel to model default/death time, we then consider three types of defaultable contingent payoffs. By allowing the survival conditioning time to be anytime before the start time of the payoffs, between the start time and end time, or after the end time of the payoffs, we provide the complete treatment of three types of defaultable contingent payoffs. As the application of the general results developed in this paper, we also provide the more general results for three types of defaultable contingent payoffs than the ones in the literature under the stochastic intensity framework.