Trajectorial Martingales and Worst Case Approach to Market Models

轨迹鞅和市场模型的最坏情况方法

• 批准号: RGPIN-2018-03867
• 负责人: Ferrando, Sebastian
• 金额: $ 1.46万
• 依托单位: Ryerson University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=650670
• 关键词: Trajectorial; Martingales; Worst; Case; Approach

Financial markets are an essential part of modern societies; they provide the needed capital to make a myriad of economical activities possible. Regrettably, the negative side of financial speculation as well as the availability of cheap and unlimited credit create undesirable side effects (e.g. market crashes, uncontrollable debt, etc). In short, there is a definite need for a better understanding of financial markets. ******Stochastic modeling in financial mathematics is presently being generalized to incorporate higher levels of uncertainty. This initiative reflects the current inability of models to incorporate all variables that affect market conditions as well as a reliable joint probability distribution. Along this modern line of research, we propose an approach that weakens dramatically basic stochastic modeling assumptions. Conclusions in such an approach are more robust as they are less dependent on prior modeling assumptions. Our work will concentrate on a set of trajectories that replace the path space of the stochastic process. A thorough investigation will be pursued of several fundamental mathematical constructions that are available in this setting without any prior probabilistic assumptions.******A trajectorial version of the fundamental notion of financial arbitrage offers the possibility to define a general notion of trajectorial martingale. The latter concept is the analogue of martingale processes and as such is poised to play a central role in our approach. Among many developments, we will construct a pathwise version of conditional expectations and will study the possibility to develop a trajectorial analogue of the different variants of pathwise stochastic integrals.******A number of fundamental analytical developments are possible in the proposed framework. In particular, there is a natural integration operator defined by superhedging, a financial based approximation that provides coverage under each eventuality (trajectory wise) and which is not associated to a classical (Kolmogorov-type) probability measure.******Our approach pays attention to individual trajectories and, as such, could also be labelled worst case. The latter concept gains relevance and specificity in each particular application where assuming an apriori probability distribution is unwarranted. Our proposal does not make any assumptions on probability distributions (i.e. measure) but explores basic results that can be obtained prior to introducing a measure. This allows to gain a conceptual understanding of the financial meaning of new and established mathematical results that are obtained in our framework as they are interpreted without the language of probabilities. We will explore the financial implications of our new conceptual approach and will propose several market model constructions.

金融市场是现代社会的重要组成部分；它们提供了所需的资本，使无数的经济活动成为可能。令人遗憾的是，金融投机的负面影响以及廉价和无限信贷的可获得性造成了不良的副作用(如市场崩溃、无法控制的债务等)。简而言之，确实有必要更好地了解金融市场。*金融数学中的随机建模目前正在被推广，以纳入更高水平的不确定性。这一举措反映了目前模型无法纳入影响市场状况的所有变量以及可靠的联合概率分布。沿着这条现代研究路线，我们提出了一种方法，极大地削弱了基本的随机建模假设。这种方法的结论更稳健，因为它们较少依赖于先前的建模假设。我们的工作将集中在一组取代随机过程的路径空间的轨迹上。在没有任何先验概率假设的情况下，我们将对几种基本的数学结构进行彻底的调查。*金融套利基本概念的轨迹版本提供了定义轨迹鞅的一般概念的可能性。后一种概念类似于鞅过程，因此，它将在我们的方法中发挥核心作用。在许多发展中，我们将构建条件期望的路径版本，并将研究开发路径随机积分的不同变体的轨迹模拟的可能性。*在建议的框架中可能有一些基本的分析发展。特别是，有一个由超级对冲定义的自然积分算子，这是一种基于金融的近似，它在每个可能性(轨迹方面)下提供覆盖，并且与经典的(Kolmogorov类型)概率度量无关。*我们的方法关注单个轨迹，因此也可以被贴上最坏情况的标签。后一种概念在假设先验概率分布是不必要的情况下，在每个特定的应用中获得相关性和特殊性。我们的建议不对概率分布(即度量)做出任何假设，而是探索在引入度量之前可以获得的基本结果。这使得我们能够从概念上理解在我们的框架中获得的新的和已建立的数学结果的财务含义，因为它们在没有概率语言的情况下被解释。我们将探索我们新概念方法的财务影响，并将提出几个市场模型构建。