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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=677333
• 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；他们提供了必要的资金，以使无数的经济活动成为可能。遗憾的是，财务投机的负面影响以及廉价和无限信贷的可用性会产生不良的副作用（例如市场崩溃，无法控制的债务等）。 简而言之，一定需要更好地了解金融市场。 ******金融数学中的随机建模目前已被概括，以纳入更高水平的不确定性。 该计划反映了模型目前无法合并影响市场条件以及可靠的联合概率分布的所有变量。沿着这一现代研究，我们提出了一种削弱基本的随机建模假设的方法。结论在这种方法中的结论更加可靠，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轨迹上，以取代随机过程的路径空间。 将对在本环境中可用的几种基本数学结构进行彻底的调查，而无需任何先前的概率假设。 。后一个概念是Martingale过程的类似物，因此有望在我们的方法中发挥核心作用。在许多发展中，我们将构建有条件期望的路径版本，并将研究开发路径随机积分不同变体的轨迹类似物的可能性。 。特别是，有一个自然集成运算符，该操作员是由超固定的定义的，这是一个基于财务的近似值，可提供每个可能性下的覆盖范围（轨迹明智），并且与经典（kolmogorov-type）概率无关。***********我们的我们方法会注意各个轨迹，因此也可能被标记为最坏情况。后一个概念在每个特定应用中都获得了相关性和特异性，因为假设Apriori概率分布是没有根据的。我们的建议没有对概率分布（即量度）做出任何假设，而是探索在引入量度之前可以获得的基本结果。这允许对新框架中获得的新数学结果的财务意义获得概念性理解，因为它们在没有概率的语言的情况下被解释了。我们将探讨我们新概念方法的财务影响，并提出几种市场模型构建。