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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758356
• 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一个概率空间，配备过滤，作为一个不折扣的Sigma-Elgebras家族。当过滤是完整且直率的，随机过程的发达理论假设“通常的条件”。该理论在1975年的概率理论，统计学，数学金融等方面产生了许多重要的结果，在1975年，随机过程中的著名专家doob和dellacherie启动了对随机过程的研究，而没有这种技术假设。 Lepingle，Horowitz，Lenglart以及Galtchouk进行了进一步的发展。他们的随机分析的并行版本涉及可选的过程，该过程允许常规轨迹。这种初始理论的存在要求为其当今的进一步发展和应用程序采取新的倡议，并面临新的挑战。该提案的目的是对可选过程进行新的了解，从而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，我们将在强溶液的存在及其路径比较属性方面研究随机分化方程。结果将应用于可选流程驱动的市场中近似期权价格界限和其他财务量。还将在使用正交多项式的股票回报概率分布的概率分布和pade Rational近似值的帮助下研究近似定价的问题。我们将在合并和收购方面的期权定价问题中使用可选过程的技术，在这些方面，跳跃过程有望创建适当的定价模型。我们研究了获取可选超级马特林群岛均匀Doob-Meyer分解版本的可能性。由于在不完整的市场，具有交易成本和其他市场限制的市场中，其在数学融资中的基本作用是建立在数学金融中的。我们还想展示如何利用这种分解来在可选的半木制的过滤问题中构建最佳过滤器，该滤波器涵盖了许多众所周知的模型。除了可选分解外，我们还将推出可选的Martingales的Galtchouk-Kunita-Watanabe代表的版本，并进一步应用于刻薄的方差对冲问题。数学金融中称为内部交易的另一个基本问题将根据可选过程的计算进行处理。将研究可选半明星的参数估计问题。这些结果是合理的，可以在可选过程驱动的市场中提供足够的校准，并为在数学金融和统计中利用的许多回归模型创建一般框架。该提案足够广泛，可以容纳许多硕士和博士学位水平的学生研究项目。