Modern Stochastics: Optional Processes and their Applications

现代随机分析的基石是一个配备过滤的概率空间，它是一个非递减的sigma-代数族。成熟的随机过程理论假定过滤是完全且右连续的“通常情况”。这一理论在概率论、统计学、数学金融学等领域产生了许多重要的结果。1975年，著名的随机过程专家Doob和Dellacherie在没有这一技术假设的情况下开始了对随机过程的研究。莱平格尔、霍洛维茨、伦格拉特，主要是加尔特楚克进行了进一步的开发。他们的平行版本的随机分析处理的是允许规则轨迹的可选过程。这种初始理论的存在呼唤着一种新的倡议，在它面临新的挑战的今天，它的进一步发展和应用。该提案的目标是以新的视角看待可选过程，为数学金融及相关领域带来新的方法、技术和结果。在该方案中，我们将研究关于可选半鞅的随机微分方程在强解的存在意义下及其路径比较性质。结果将被应用于由可选过程驱动的市场中近似的期权价格界限和其他金融数量。本文还将利用多项式和Pade有理逼近对股票收益率的概率分布进行扩展，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的技术，其中跳跃过程承诺创建一个适当的定价模型。我们研究了得到可选上鞅的一致Doob-Meyer分解的一种形式的可能性。由于它在不完全市场、有交易成本的市场和其他市场限制下的超级套期保值问题中的应用，它在数学金融中的基础作用是根深蒂固的。我们还想展示如何利用这种分解在覆盖许多著名模型的可选半鞅的过滤问题中构造最优过滤器。除了可选分解外，我们还将得到可选鞅的Galtchouk-Kurita-Watanabe表示的一个版本，并进一步应用于均值-方差套期保值问题。数学金融学中另一个被称为内幕交易的基本问题将基于可选过程的演算进行处理。研究了可选半鞅的参数估计问题。这些结果是合理的，可以在由可选过程驱动的市场中提供适当的校准，并为数学、金融和统计中使用的许多回归模型创建一个通用框架。该提案的范围足够广泛，足以容纳一些硕士和博士水平的学生研究项目。

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.