Modern Stochastics: Optional Processes and their Applications

现代随机指标：可选过程及其应用

• 批准号: RGPIN-2019-04922
• 负责人: Melnikov, Alexander
• 金额: $ 1.46万
• 依托单位: University of Alberta
• 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=715234
• 关键词: Modern; Stochastics; Optional; Processes; their

The cornerstone of modern stochastic analysis is a probability space equipped with filtration as a non-decreasing family of sigma-algebras. The well-developed theory of stochastic processes assumes “usual conditions”, when filtration is complete and right-continuous. This theory generated many important results in probability theory, statistics, mathematical finance etc. In 1975 famous experts in stochastic processes Doob and Dellacherie initiated studies of stochastic processes without this technical assumption. Further developments were done by Lepingle, Horowitz, Lenglart, and mostly by Galtchouk. Their parallel version of stochastic analysis deals with optional processes admitting regular trajectories. The existence of such initial theory calls for a new initiative for its further developments and applications today with its new challenges. The goal of the proposal is to take a new look at optional processes bringing new methods, techniques and results to mathematical finance and related areas. In the proposal, we are going to investigate stochastic differential equations with respect to optional semimartingales in the sense of existence of strong solutions and their path-wise comparison properties. The results will be applied to approximate option price bounds and other financial quantities in the markets driven by optional processes. The problem of approximate pricing will be also investigated with the help of extensions of probability distributions of stock returns using orthogonal polynomials and the Pade rational approximations. We are going to use the technique of optional processes in option pricing problem in the area of mergers and acquisitions, where jump processes promise to create an adequate pricing model. We investigate a possibility to obtain a version of the uniform Doob-Meyer decomposition of optional supermartingales. Its fundamental role in mathematical finance is well-established due to its application to superhedging problem in incomplete markets, markets with transaction costs and other market restrictions. We also want to show how this decomposition can be exploited to construct an optimal filter in the filtering problem for optional semimartingales which covers many well-known models. Besides optional decomposition, we will derive a version of the Galtchouk-Kunita-Watanabe representation for optional martingales with further applications to mean-variance hedging problem. Another fundamental problem known in mathematical finance as insider trading will be treated based on the calculus of optional processes. The parameter estimation problem for optional semimartingales will be investigated. These results are reasonable to provide an adequate calibration in the markets driven by optional processes and to create a general framework for many regression models exploited in mathematical finance and statistics. The Proposal is wide enough to accommodate a number of student research projects of master's and PhD levels.

现代随机分析的基石是配备有过滤功能的概率空间，作为非减西格玛代数族。成熟的随机过程理论假设“通常条件”，即过滤是完全且右连续的。该理论在概率论、统计学、数理金融等领域产生了许多重要成果。1975年，著名随机过程专家Doob和Dellacherie发起了没有这一技术假设的随机过程的研究。进一步的开发由 Lepingle、Horowitz、Lenglart 以及主要由 Galtchouk 完成。他们的随机分析的并行版本涉及允许规则轨迹的可选过程。这种初始理论的存在需要新的举措来进一步发展和应用，并面临新的挑战。该提案的目标是重新审视可选流程，为数学金融及相关领域带来新的方法、技术和结果。在该提案中，我们将在强解的存在性及其路径比较属性的意义上研究关于可选半鞅的随机微分方程。结果将应用于近似期权价格范围和由可选流程驱动的市场中的其他金融数量。还将借助使用正交多项式和 Pade 有理近似的股票收益概率分布扩展来研究近似定价问题。我们将在并购领域的期权定价问题中使用可选过程技术，其中跳跃过程有望创建一个适当的定价模型。我们研究了获得可选超鞅的统一 Doob-Meyer 分解版本的可能性。由于其应用于不完全市场、具有交易成本的市场和其他市场限制的超级对冲问题，其在数学金融中的基本作用已得到确立。我们还想展示如何利用这种分解来构建可选半鞅的过滤问题中的最优过滤器，其中涵盖了许多众所周知的模型。除了可选分解之外，我们还将导出可选鞅的 Galtchouk-Kunita-Watanabe 表示的一个版本，并进一步应用于均值方差对冲问题。数学金融中称为内幕交易的另一个基本问题将根据可选过程的计算来处理。将研究可选半鞅的参数估计问题。这些结果是合理的，可以为可选流程驱动的市场提供充分的校准，并为数学金融和统计学中使用的许多回归模型创建通用框架。该提案的范围足够广泛，可以容纳许多硕士和博士级别的学生研究项目。