We consider a financial framework with two levels of information: the public information generated by the financial assets, and a larger flow of information that contains additional knowledge about a random time. This random time can represent many economic and financial settings, such as the default time of a firm for credit risk, and the death time of an insured for life insurance. As the random time cannot be seen before its occurrence, the progressive enlargement of filtration seems tailor‐fit to model the larger flow of information that incorporates both the public flow and the information about the random time. In this context, our interest focuses on the following challenges: (a) How to single out the various risks coming from the financial assets, the random time, and their correlations? (b) How these risks interplay and lead to the formation of any risk in the larger flow of information? It is clear that understanding how risks build‐up and interact, when one enlarges the flow of information, is vital for an efficient risk management and derivatives' evaluation in those informational markets. Our answers to these challenges are full and complete no matter what the model for the random time is and no matter how the random time is related to the public flow. In fact, we introduce “pure default” risks, and quantify and classify these risks afterward. Then we elaborate our martingale representation results, which state that any martingale in the large filtration stopped at the random time can be decomposed into orthogonal local martingales (i.e., local martingales whose product remains a local martingale). This constitutes our first principal contribution, while our second contribution consists in evaluating various defaultable securities according to the recovery policy, within our financial setting that encompasses any default model, using a martingale “basis.” Our pricing formulas explain the impact of various recovery policies on securities and determine the types of pure default risk they entail.

中文翻译：

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ting表示定理和可违约证券的估值

我们考虑具有两个信息级别的金融框架：由金融资产生成的公共信息，以及包含有关随机时间的其他知识的较大信息流。这个随机时间可以代表许多经济和金融环境，例如公司信用风险的默认时间以及人寿保险的死亡时间。由于无法在随机时间发生之前看到随机时间，因此过滤的逐步扩大似乎是量身定制的模型，可以对包含公共流量和随机时间信息的较大信息流进行建模。在这种情况下，我们的兴趣集中在以下挑战上：（a）如何挑选出金融资产，随机时间，及其相关性？（b）这些风险如何相互作用，并导致在更大的信息流中形成任何风险？显然，了解风险如何在信息流增加时建立和相互作用，对于在这些信息市场中进行有效的风险管理和衍生产品评估至关重要。无论随机时间的模型是什么，以及随机时间与公众流量之间的关系如何，我们对这些挑战的回答都是完整而完整的。实际上，我们引入了“纯违约”风险，然后对这些风险进行量化和分类。然后，我们详细阐述mar的表示结果，该结果表明，在随机时间停止的大过滤中的任何mar都可以分解为正交的本地mar（即，其乘积仍为本地mar的本地mar）。这构成了我们的第一个主要贡献，而我们的第二个贡献在于，在包括任何违约模型在内的我们的财务环境中，使用a“基础”根据回收政策评估各种违约证券。我们的定价公式解释了各种回收政策对证券的影响，并确定了它们所带来的纯违约风险的类型。