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摘要翻译：
我们考虑了一个或多个公司违约时间建模的基于强度的方法。在该方法中，默认时间定义为Cox过程的跳跃时间，该过程是以实现强度为条件的泊松过程。我们假设强度遵循Cox-Ingersoll-Ross模型。该模型允许人们明确地计算违约债券的生存概率和价格。在本文中，我们假定没有观察到驱动强度的布朗运动。利用点过程观测的滤波理论，我们可以导出强度及其矩母函数的动力学，给定Cox过程的观测结果。条件矩母函数的动力学变换允许我们解决Cox过程跳变之间以及跳变处的滤波问题。假设光强的初始分布为Gamma型，对于所有t>0，我们得到了滤波问题的显式解。我们通过观察得到的条件矩母函数在时间t对应于混合的伽马分布来总结这篇论文。
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英文标题：
《Explicit Computations for a Filtering Problem with Point Process
Observations with Applications to Credit Risk》
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作者：
Vincent Leijdekker and Peter Spreij
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最新提交年份：
2008
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分类信息：
一级分类：Quantitative Finance 数量金融学
二级分类：Computational Finance 计算金融学
分类描述：Computational methods, including Monte Carlo, PDE, lattice and other numerical methods with applications to financial modeling
计算方法，包括蒙特卡罗，偏微分方程，格子和其他数值方法，并应用于金融建模
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一级分类：Mathematics 数学
二级分类：Probability 概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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英文摘要：
We consider 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, which is a Poisson process conditional on the realization of its intensity. We assume that the intensity follows the Cox-Ingersoll-Ross model. This model allows one to calculate survival probabilities and prices of defaultable bonds explicitly. In this paper we assume that the Brownian motion, that drives the intensity, is not observed. Using filtering theory for point process observations, we are able to derive 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 the filtering problem, between the jumps of the Cox process, as well as at the jumps. 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 paper with the observation that the resulting conditional moment generating function at time t corresponds to a mixture of Gamma distributions.
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PDF链接：
https://arxiv.org/pdf/0802.1407
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