英文标题：
《Quantum Duality in Mathematical Finance》
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作者：
Paul McCloud
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最新提交年份：
2017
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英文摘要：
  Mathematical finance explores the consistency relationships between the prices of securities imposed by elementary economic principles. Commonplace among these are replicability and the absence of arbitrage, both essentially algebraic constraints on the valuation map from a security to its price.   The discussion is framed in terms of observables, the securities, and states, the linear and positive maps from security to price. Founded on the principles of replicability and the absence of arbitrage, mathematical finance then equates to the theory of positive linear maps and their numeraire invariances. This acknowledges the algebraic nature of the defining principles which, crucially, may be applied in the context of quantum probability as well as the more familiar classical setting.   Quantum groups are here defined to be dual pairs of *-Hopf algebras, and the central claim of this thesis is that the model for the dynamics of information relies solely on the quantum group properties of observables and states, as demonstrated by the application to finance. This naturally leads to the study of models based on restrictions of the *-Hopf algebras, such as the Quadratic Gauss model, that retain much of the phenomenology of their parent within a more tractable domain, and extensions of the *-Hopf algebras, such as the Linear Dirac model, with novel features unattainable in the classical case.
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中文摘要：
数学金融学探索由基本经济原则施加的证券价格之间的一致性关系。其中最常见的是可复制性和无套利，这两个基本上都是从证券到其价格的估值图上的代数约束。讨论的框架是可观测的、证券和状态，以及从证券到价格的线性和正映射。数学金融学建立在可复制性和无套利原则的基础上，然后等同于正线性映射及其数字不变性的理论。这承认了定义原则的代数性质，至关重要的是，这些原则可以应用于量子概率以及更熟悉的经典背景中。量子群在这里被定义为*-Hopf代数的对偶对，本论文的中心主张是，信息动力学模型完全依赖于可观测和状态的量子群属性，如金融应用所示。这自然会导致对基于*-霍普夫代数限制的模型的研究，例如二次高斯模型，该模型将其父代的许多现象学保留在一个更容易处理的领域内，以及对*-霍普夫代数的扩展，例如线性狄拉克模型，具有经典情况下无法实现的新特性。
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Mathematical Finance        数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
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一级分类：Mathematics        数学
二级分类：Quantum Algebra        量子代数
分类描述：Quantum groups, skein theories, operadic and diagrammatic algebra, quantum field theory
量子群，skein理论，运算代数和图解代数，量子场论
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