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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=712297
• 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；它们提供了使无数经济活动成为可能所需的资本。遗憾的是，金融投机的负面影响以及廉价且无限的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，确实需要更好地了解金融市场。 金融数学中的随机模型目前正在推广以纳入更高水平的不确定性。 这一举措反映了当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。沿着这一现代研究路线，我们提出了一种显着削弱基本随机建模假设的方法。这种方法的结论更加稳健，因为它们较少依赖于先前的建模假设。 我们的工作将集中于一组替代随机过程路径空间的轨迹。 将对在这种情况下可用的几个基本数学结构进行彻底的调查，而无需任何事先的概率假设。 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。后一个概念类似于鞅过程，因此将在我们的方法中发挥核心作用。在众多进展中，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。 在拟议的框架中，许多基本的分析发展是可能的。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，它提供了每种可能性（轨迹方面）的覆盖范围，并且与经典（柯尔莫哥洛夫型）概率度量无关。 我们的方法关注个体轨迹，因此也可以被标记为最坏情况。后一个概念在每个特定应用中都具有相关性和特异性，其中假设先验概率分布是没有根据的。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。这使得我们能够对在我们的框架中获得的新的和已建立的数学结果的财务意义有一个概念性的理解，因为它们是在没有概率语言的情况下解释的。我们将探讨新概念方法的财务影响，并提出几种市场模型构建。