Non-Additive Criterion on Controlled Markov Chains and its Applications to Mathematical Finance

受控马尔可夫链的非可加性准则及其在数学金融中的应用

基本信息

批准号：
13440036
负责人：
IWAMOTO Seiichi
金额：
$ 5.76万
依托单位：
Kyushu University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2004
项目状态：
已结题

项目摘要

This project studies from a viewpoint of non-additivity, which is well-known as non-linearity in criterion. Our aim is to clarify an optimal structure both in policy and in system dynamics under the expected utility criteria. Further we also apply the results we have obtained for optimization scheme to non-optimization problems such as option evaluation in mathematical finance.It has been well known that dynamic optimizations such as in Markov decision processes have treated additive (linear) criteria e.g. discounted total expected reward. These optimizations are relatively easily performed because of linearity in expectation operator: However what will happen when we optimize such non-additive criteria? Our study begins with abandoning the linearity in expectation. Instead, we focus our attention to (a)monotonicity in expectation operator, (b)associativity in reward criteria and (c)successive applicability of state dynamics.By using these three properties, we have succeeded in establi … More shing a large variety of optimization methods and in recognizing that these dynamic methods turn out to be fruitful in implemnentation/calculation. To be more concretely specific, we have obtained the following four results.(1)To have introduced policy classes and classified them into Markov, general, primitive and expanded Markov,(2)To have introduced two criterion classes and classified both into simple criterion (additive, multiplicative, maximum and terminal) and compound criterion (range, variance, ratio, sum excluding extrema and mid-range),(3)To have associated the policy classes with the two criterion classes and presented how to imbed the original problem into an expanded class of problems,(4)To have applied the results obtained for wide class of optimization problems in (1)-(3) to a class of non-optimization problems in mathematical finance such as option pricing.Throughout this study, we have clarified that a large class of optimization methods have been useful for a wide variety of criteria. These methods have been restricted to a small class of deterministic problems. Now our approach has shown that these methods are applicable to a huge class of optimization and non-optimization problems in stochastic, fuzzy or non-deterministic problems. In particular we have developed an evaluation method of a large class of options (derivatives) in mathematical finance. This is called dynamic pricing or recursive evaluation. Further more we have succeeded in developing some new derivatives such as options with random expiration date…Pacific options, look back American options and others.As a summary we have made dynamic optimization method fiuitful Thus dynamic programming, recursive method, and invariant imbedding turn out to be useful for a wide class of optimization and/or evaluation problems which inherits (i)non-additivity, (ii)non-linearity or (iii)non-determinancy. This project has just opened the door to overcome the three difficulties. Less

该项目从政策和预期的效用标准中的h的观点进行了研究。在Markison过程中，由于期望运算符的线性，可以进行添加的（线性）标准。（a）投资操作员的单调性，（b）奖励标准中的关联性和（c）状态动力学的连续适用性。通过使用三个这些替代品，建立...更多地提出了一种野生的优化方法，并认识到thamic方法转弯在离子中是富有成效的，我们已经获得了能量，原始和扩展的马尔可夫，（2）已引入两个标准类，并将其分类为简单的标准复合标准（范围已将策略类别与两个标准类相关联，并介绍了如何将原始问题嵌入到扩展的一系列问题中，（4）已应用（1）优化问题中获得的广泛优化问题获得的结果在数学财务中，我们有多种优化方法对各种各样的标准有用。 - 尤其是在较大的选项中，我们已经开发了一些新的衍生品评估。日期…太平洋选项，回顾美国的选项。作为摘要，我们具有DE DYAMIC优化方法，因此动态编程，递归方法和不变性嵌入材料可用于继承的广泛优化和/或评估问题（I ）非添加性，非线性或（iii）只是打开了较少的差异