Informational markets: Risk modelling, risk management and portfolio analysis

信息市场是一个有两种信息流的市场。用F表示的一个流代表“公共”信息，它对所有代理人都是可用的，还有一个更大的流G，它包含F和有关可能影响市场的事件的附加信息。因此，我们在这个项目中的出发点是由（S，F）对描述的初始市场模型，我们添加了几个随机时间。这里S代表股票的价格过程，F是对信息“公开”进行建模的过滤。随机时间可以模拟人寿保险的投保代理人的死亡时间，或者信用风险理论的公司违约时间，或者更一般地说，可能影响市场的事件的发生时间。鉴于死亡时间在发生之前无法看到，用于建模F，G和随机时间之间的连接的逐步放大方法似乎适合于上述领域。在这种情况下，我们的主要目标在于理解由随机时间中的不确定性单独和/或与过滤F共同引起的不同类型的风险。准确地说，我们有兴趣回答以下问题：我们能挑出可以归类为“纯随机时间”风险的风险吗？有多少这样的风险？这些风险如何与来自初始市场（S，F）的风险相互作用？最重要的是，我们是否可以为较大的流G定义一个风险基，并且关于这个过滤的每个风险都可以关于这个基唯一地分解？通过对这些问题的详细回答，我们将深刻理解风险是如何积累的，只要我们在模型中加入更多的不确定性，我们就为风险提供了一个前沿的数学建模和量化。这将使我们能够有效地设计风险管理工具。除此之外，我们研究的预期应用是多种多样的。首先，我们将能够充分了解额外的不确定性在多大程度上影响了套利理论的基本金融和/或经济概念以及与市场流动性密切相关的市场生存能力。此外，当这些概念对（S，G）失效时，我们应该能够建议市场的监管政策，以修复由“额外”信息引起的损害。其次，我们要描述的动态和随机结构的死亡率和/或长寿衍生品，如长寿债券，这是非常重要的死亡率/长寿证券化。我们也有兴趣完整地描述G下所有平减指数的集合，除了在投资组合分析中的重要作用之外，它本身也很有趣。最后，我们将研究随机时间对最优投资组合（对于各种优化标准）、计价单位投资组合以及现有定价规则的影响。

An informational market is a market where there are two flows of information. One flow denoted by F represents the ``public" information, which is available to all agents, and a larger flow G that contains F and additional information about events that might affect the market. Thus, our starting point in this project is the initial market model described by the pair (S,F), to which we add several random times. Here S represents the stocks' price process and F is the filtration that models the ``public" of information. The random times can model the death times of insured agents for life insurance, or the defaults times of firms for credit risk theory, or more generally the occurrence times of events that might impact the market. Given that death times can not be seen before they occur, the progressively enlargement approach for modelling the connection between F, G and the random times, seems tailored-fit for these aforementioned domains. In this setting, our main objectives reside in understanding the different types of risks induced by the uncertainties in the random times alone and/or jointly with the filtration F. Precisely, we are interested in answering the following questions: Can we single out the risk(s) that can be classified as ``pure random times' risk"? How many are there of such risk(s)? How these risk(s) interplay with the risk(s) coming from the initial market (S,F)? Above all, can we define a risk-basis for the larger flow G, and every risk with respect to this filtration can be decomposed uniquely with respect to this basis? Via detailed answers to these questions, we will deeply understand how risks build up as soon as we add more uncertainties to a model, we provide a cutting- edge mathematical modelling and quantification for the risk. This will allow us to efficiently design risk management tools afterwards. Besides these, the projected applications of our study are numerous and multifold. First of all, we will be able to tell fully how far the additional uncertainties affect the fundamental financial and/or economic concepts of arbitrage theory and market's viability that are intimately related to market's liquidity. Furthermore, when these concepts fail for (S,G), we should be able to suggest market's regulation(s) policies in order to repair the damage induced by the ``extra" information. Secondly, we want to describe the dynamics and the stochastic structures for mortality and/or longevity derivatives, such as longevity bonds, which are highly important for the mortality/longevity securitization. We are also interested in describing completely and fully the set of all deflators under G, which is interesting in itself besides its vital role in the portfolio analysis. Finally, we are will investigate the impact of the random times on optimal portfolios (for various optimization criterion), the numeraire portfolio, and on the existing pricing rules as well.