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

本计画从非加性的观点来研究，也就是所谓的非线性准则。我们的目标是澄清一个最佳的结构，在政策和系统动力学下的期望效用准则。此外，我们也适用于我们所获得的结果，优化方案的非优化问题，如在数学financy.It的期权评估已被公知的动态优化，如在马尔可夫决策过程处理添加剂（线性）的标准，如贴现总期望回报。由于期望算子的线性，这些优化相对容易执行：然而，当我们优化这种非可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，我们把注意力集中在（a）期望算子的单调性，（B）奖励准则的结合性和（c）状态动力学的连续适用性上。 ...更多信息 我们拥有大量的优化方法，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)To引入了策略类，并将策略类分为马尔可夫类、一般马尔可夫类、原始马尔可夫类和扩展马尔可夫类;（2）引入了两个准则类，并将准则类分为简单准则类（加法、乘法、最大值和终结值）和复合准则（范围、方差、比率、总和，不包括极值和中间值），（3）将策略类与两个准则类相关联，提出了如何将原问题嵌入到扩展问题类中，（4）将（1）-（3）中对广泛的优化问题所得到的结果应用于数学金融中的一类非优化问题，如期权定价，在整个研究中，我们已经阐明了一大类优化方法对各种各样的准则都是有用的。这些方法仅限于一类确定性问题。现在，我们的方法已经表明，这些方法适用于一个巨大的一类随机，模糊或不确定性问题的优化和非优化问题。特别是，我们已经开发了一个评估方法的一大类期权（衍生品）的数学金融。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工具，如具有随机到期日的期权、太平洋期权、回望美式期权等。作为一个总结，我们使动态优化方法富有成效。因此，动态规划、递归方法和不变嵌入对一类广泛的优化和/或评估问题是有用的，这些问题继承了（i）非可加性，（ii）非线性或（iii）不确定性。这个项目刚刚打开了克服三大困难的大门。少

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