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EXPLICIT COMPUTATIONS FOR A FILTERING PROBLEM WITH POINT PROCESS OBSERVATIONS WITH APPLICATIONS TO CREDIT RISK

Published online by Cambridge University Press:  17 May 2011

Vincent Leijdekker  and
Peter Spreij
Vincent Leijdekker
Affiliation:
Product Analysis, ABN AMRO Bank N.V., Gustav Mahlerlaan 10, 1082 PP Amsterdam, The Netherlands E-mail: vincent.leijdekker@gmail.com
Peter Spreij
Affiliation:
Korteweg-de Vries Institute for Mathematics, Universiteit van Amsterdam, PO Box 94248, 1090GE Amsterdam, The Netherlands E-mail: spreij@uva.nl
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Abstract

We consider the filtering problem for a doubly stochastic Poisson or Cox process, where the intensity follows the Cox–Ingersoll–Ross model. In this article we assume that the Brownian motion, which drives the intensity, is not observed. Using filtering theory for point process observations, we first derive the dynamics for the intensity and its moment-generating function, given the observations of the Cox process. A transformation of the dynamics of the conditional moment-generating function allows us to solve in closed form the filtering problem, between the jumps of the Cox process as well as at the jumps, which constitutes the main contribution of the article. Assuming that the initial distribution of the intensity is of the Gamma type, we obtain an explicit solution to the filtering problem for all t>0. We conclude the article with the observation that the resulting conditional moment-generating function at time t, after Nt jumps, corresponds to a mixture of Nt+1 Gamma distributions. Currently, the model that we analyze has become popular in credit risk modeling, where one uses the intensity-based approach for the modeling of default times of one or more companies. In this approach, the default times are defined as the jump times of a Cox process. In such a model, one only has access to observations of the Cox process, and thus filtering comes in as a natural technique in credit risk modeling.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 25 , Issue 3 , July 2011 , pp. 393 - 418
Copyright
Copyright © Cambridge University Press 2011

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