Modern Stochastics: Optional Processes and their Applications

现代随机分析的基石是一个概率空间配备了过滤作为一个非减家庭的西格玛代数。发展完善的随机过程理论假设“通常条件”，即过滤是完全的和右连续的。这一理论产生了许多重要成果的概率论，统计学，金融数学等在1975年著名专家在随机过程Doob和Dellacherie发起研究随机过程没有这种技术假设。进一步的发展是由Lepingle，Horowitz，Lenglart完成的，主要是Galtchouk。他们的平行版本的随机分析处理的可选过程承认定期的轨迹。这一初始理论的存在，要求在新的挑战下，为它的进一步发展和应用提出新的倡议。该提案的目标是重新审视可选过程，为数学金融和相关领域带来新的方法，技术和结果。在这个建议中，我们将研究随机微分方程关于任选半鞅的强解的存在性及其路径比较性质。结果将被应用到近似的期权价格边界和其他金融量的市场驱动的可选过程。近似定价的问题也将调查的帮助下，使用正交多项式和帕德有理逼近的股票收益率的概率分布的扩展。我们将在兼并和收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。本文研究了任意上鞅的一致Doob-Meyer分解的一种可能性。它在数学金融中的基础作用是公认的，因为它应用于不完全市场，有交易费用的市场和其他市场限制的超套期保值问题。我们还想展示如何利用这种分解来构建一个最佳的过滤器的过滤问题的可选半鞅，其中包括许多著名的模型。除了可选分解，我们将推出一个版本的Galtchouk-Kunita-Watanabe表示可选鞅进一步应用到均值-方差套期保值问题。数学金融学中的另一个基本问题是内幕交易，我们将基于可选过程的微积分来处理。研究了可选半鞅的参数估计问题。这些结果是合理的，提供了一个适当的校准，在市场驱动的可选过程，并创建一个一般框架，许多回归模型在数学金融和统计。该提案足够广泛，可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.