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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

本课题从非可加性的角度进行研究，而非可加性在判据中被称为非线性。我们的目标是在预期效用标准下，阐明政策和系统动力学的最佳结构。进一步将优化方案的结果应用于数学金融中的期权评估等非优化问题。众所周知，动态优化，如马尔可夫决策过程，已经处理了加性（线性）标准，如贴现总预期奖励。由于期望操作符的线性，这些优化相对容易执行：但是，当我们优化这些非加性标准时会发生什么？我们的研究从放弃期望的线性开始。相反，我们将注意力集中在(a)期望算子的单调性，(b)奖励标准的结合性和(c)状态动力学的连续适用性。通过使用这三个特性，我们成功地建立了各种各样的优化方法，并认识到这些动态方法在实现/计算中是富有成效的。更具体地说，我们得到了以下四个结果。(1)引入了策略类，并将其分为马尔可夫、一般、原始和扩展马尔可夫；(2)引入了两个准则类，并将其分为简单准则（相加性、乘法性、最大值和终末性）和复合准则(范围、方差、比率、(3)将策略类与两个准则类联系起来，并提出如何将原始问题嵌入到扩展的问题类中，(4)将(1)-(3)中广泛的优化问题类的结果应用于数学金融中的一类非优化问题，如期权定价。在整个研究过程中，我们已经澄清了一大类优化方法对各种各样的标准都是有用的。这些方法仅限于一小类确定性问题。现在我们的方法已经表明，这些方法适用于随机、模糊或不确定性问题中的大量优化和非优化问题。特别是，我们已经开发了一种评估方法，在数学金融的期权（衍生品）大类。这被称为动态定价或递归评估。此外，我们还成功地开发了一些新的衍生品，如随机到期日期权、太平洋期权、回溯美国期权等。综上所述，我们已经使动态优化方法有效，因此动态规划，递归方法和不变嵌入对于继承(i)非可加性，（ii）非线性或（iii）非确定性的广泛的优化和/或评估问题是有用的。这个项目正好打开了克服这三个困难的大门。少