英文标题：
《A martingale representation theorem and valuation of defaultable
  securities》
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作者：
Tahir Choulli, Catherine Daveloose, Mich\\`ele Vanmaele
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最新提交年份：
2018
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英文摘要：
  We consider a market model where there are 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. By using the expansion of filtration, the random time uncertainty and its entailed risk are fully considered without any mathematical restriction. In this context with no model\'s specification for the random time, the main challenge lies in finding the dynamics and the structures for the value processes of defaultable or mortality and/or longevity securities which are vital for the insurance securitization. To overcome this obstacle, we elaborate our optional martingale representation results, which state that any martingale in the large filtration stopped at the random time can be decomposed into precise and unique orthogonal local martingales (i.e. local martingales whose product remains a local martingale). This constitutes our first and probably the principal contribution. Even though the driving motivation for this representation resides in credit risk theory, our results are applicable to several other financial and economics contexts, such as life insurance and financial markets with random horizon. Thanks to this optional representation, we decompose any defaultable or mortality and/or longevity liability into the sum of \"non-correlated\" risks using a risk basis. This constitutes our second contribution.
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中文摘要：
我们考虑一个市场模型，其中有两个级别的信息。金融资产产生的公共信息，以及包含随机时间额外知识的更大信息流。这种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，以及人寿保险被保险人的死亡时间。利用滤波展开，在不受任何数学约束的情况下，充分考虑了随机时间不确定性及其带来的风险。在这种情况下，由于没有对随机时间的模型说明，主要的挑战在于找到对保险证券化至关重要的可违约或死亡和/或寿命证券的价值过程的动力学和结构。为了克服这个障碍，我们详细阐述了可选的鞅表示结果，该结果表明，在随机时间停止的大过滤中的任何鞅都可以分解为精确且唯一的正交局部鞅（即乘积仍然是局部鞅的局部鞅）。这是我们的第一次，也可能是主要贡献。尽管这种表述的驱动动机存在于信用风险理论中，但我们的结果也适用于其他一些金融和经济学背景，如人寿保险和随机期金融市场。由于这种可选表示，我们使用风险基础将任何可违约或死亡和/或寿命责任分解为“非相关”风险之和。这是我们的第二个贡献。
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Mathematical Finance        数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
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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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