向量随机积分与资产定价基本定理
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摘要
本论文以向量随机积分为主线,首先较为系统地研究了向量随机积分,给出了在金融数学中使用向量随机积分较之使用按分量的随机积分更合理的经济学解释.其次作为向量随机积分的应用,研究了在金融数学中具有基础性地位的资产定价基本定理.在这两个方面都获得了一些新结果,并改进了许多已有的结论.所有这些工作不仅对于随机分析的理论,而且对于随机积分在金融数学中的应用都具有积极的意义.
本文的主要创新点如下:
1.给出了多维函数形式的单调类定理,它是一维函数形式单调类定理的推广,也是处理多维可测函数、多维随机过程的有力工具.将其用于向量随机积分的研究,改进了多个结论的证明,新证明较之原证明简洁清晰,易于掌握.
2.得到了向量随机积分一些新的性质,改进了一些已有结果.特别是利用局部绝对连续测度的密度过程、鞅空间H1与半鞅空间的性质以及泛函分析中的闭图像定理(见定理2.4.5与定理2.4.10的证明)获得了一般形式的半鞅向量随机积分的Girsa-nov定理,它对于随机分析的理论与随机积分的应用都具有重要价值.正是由于该定理,使得能够在第三章中获得新结果.
3.利用鞅空间H1的泛函表示定理、泛函延拓定理、半鞅向量随机积分的Girsanov定理(定理2.4.10)以及随机分析中许多技巧得到了半鞅的可料表示性(定理3.4.5),这一结果同样具有很好的理论与应用价值.运用它扩展了市场完备特征.
4.小结了几种不同意义下向量随机积分之间的关系,给出了一些最常用的特殊结果.还给出了等价鞅测度、等价局部鞅测度、局部绝对连续的局部鞅测度、等价鞅变换测度、等价局部鞅变换测度、局部绝对连续的局部鞅变换测度之间的关系.特别是利用本文给出的半鞅向量随机积分的Girsanov定理,得到了两个更深刻的结果(定理3.2.1与定理3.2.7) .
5.给出了怎样由一个d维可料过程获得一个d + 1维的自筹资策略的严谨证明(定理3.1.5).透彻地分析了原市场与折准市场之间、原市场的自筹资策略与折准市场的可取策略之间的关系.从数学与经济学的角度,讨论了不同形式的资产定价第一基本定理之间以及它们之间的关系.
This dissertation focuses on the vector stochastic integral (VSI) and its applicationsto mathematical finance. We first conduct a comprehensive study on VSI and provide aneconomical justification on the question of why VSI is more suitable than the component-wise stochastic integral for mathematical finance. Then as an application of VSI, we studythe fundamental theorems of asset pricing (FPAP), one of the theoretical foundations formathematical finance. Both studies generate interesting new results and improvements ofmany existing results. The work in this dissertation not only has significant contribution tothe theory of stochastic analysis, it also has very important practical value in mathematicalfinance.
The key contributions of this dissertation are summarized as follows:
1. The monotone class theorem of multiple dimension functional format.This is the generalization from the one dimension monotone class theorem. It is also apowerful tool for the study of multi-dimension measurable functions and multi-dimensionstochastic process. We apply it to the study of VSI and improve the proof of manytheorems. Comparing to the original proofs, the new proofs are significantly simplifiedand easy to understand.
2. New properties and improvements on VSI. Combining the density processof local absolute continuous measure, properties of the martingale space H1 and semi-martingale space, and the closed image theorem of functional analysis (see the proofs ofTheorem 2.4.5 and Theorem 2.4.10), we obtain the general form of Girsanov Theorem forsemi-martingale vector stochastic integral(SMVSI) . This result is of particular value tothe theory of stochastic analysis and the application of stochastic integral. It also providesthe theoretical foundation for our study in mathematical finance.
3. The semi-martingale predictable representation theorem (Theorem3.4.5). This theorem is derived from the functional representation theorem in martin-gale space H1 , the expansion theorem in functional analysis, and the Girsanov Theoremfor SMVSI (Theorem 2.4.10). Many techniques in stochastic analysis are used to obtainthis result, which also has high theoretical and practical value. Using this result, we areable to extend the characteristics of the market completeness.
4. Summary of important existing results. We give the relationships of VSI un-der several di?erent meanings and list the most commonly used special cases. More specif- ically, this includes equivalent martingale measure, equivalent local martingale measure,local absolutely continuous local martingale measure, equivalent martingale transformmeasure, equivalent local martingale transform measure, and local absolutely continuouslocal martingale transform measure. Two deeper theorems (Theorem 3.2.1 and Theorem3.2.7) are obtained from our Girsanov Theorem for SMVSI.
5. Proof of theorem on how to derive the d+1 dimension self-financingstrategy from a d-dimension predictable process (Theorem 3.1.5). After a carefulstudy of relationships between the original market and the deflated market, and theircorresponding strategies (self-financing strategy and admissible strategy, respectively),we discuss, from both mathematics and economics point of view, the first fundamentaltheorem of asset pricing in different forms and their relationship.
本文的主要创新点如下:
1.给出了多维函数形式的单调类定理,它是一维函数形式单调类定理的推广,也是处理多维可测函数、多维随机过程的有力工具.将其用于向量随机积分的研究,改进了多个结论的证明,新证明较之原证明简洁清晰,易于掌握.
2.得到了向量随机积分一些新的性质,改进了一些已有结果.特别是利用局部绝对连续测度的密度过程、鞅空间H1与半鞅空间的性质以及泛函分析中的闭图像定理(见定理2.4.5与定理2.4.10的证明)获得了一般形式的半鞅向量随机积分的Girsa-nov定理,它对于随机分析的理论与随机积分的应用都具有重要价值.正是由于该定理,使得能够在第三章中获得新结果.
3.利用鞅空间H1的泛函表示定理、泛函延拓定理、半鞅向量随机积分的Girsanov定理(定理2.4.10)以及随机分析中许多技巧得到了半鞅的可料表示性(定理3.4.5),这一结果同样具有很好的理论与应用价值.运用它扩展了市场完备特征.
4.小结了几种不同意义下向量随机积分之间的关系,给出了一些最常用的特殊结果.还给出了等价鞅测度、等价局部鞅测度、局部绝对连续的局部鞅测度、等价鞅变换测度、等价局部鞅变换测度、局部绝对连续的局部鞅变换测度之间的关系.特别是利用本文给出的半鞅向量随机积分的Girsanov定理,得到了两个更深刻的结果(定理3.2.1与定理3.2.7) .
5.给出了怎样由一个d维可料过程获得一个d + 1维的自筹资策略的严谨证明(定理3.1.5).透彻地分析了原市场与折准市场之间、原市场的自筹资策略与折准市场的可取策略之间的关系.从数学与经济学的角度,讨论了不同形式的资产定价第一基本定理之间以及它们之间的关系.
This dissertation focuses on the vector stochastic integral (VSI) and its applicationsto mathematical finance. We first conduct a comprehensive study on VSI and provide aneconomical justification on the question of why VSI is more suitable than the component-wise stochastic integral for mathematical finance. Then as an application of VSI, we studythe fundamental theorems of asset pricing (FPAP), one of the theoretical foundations formathematical finance. Both studies generate interesting new results and improvements ofmany existing results. The work in this dissertation not only has significant contribution tothe theory of stochastic analysis, it also has very important practical value in mathematicalfinance.
The key contributions of this dissertation are summarized as follows:
1. The monotone class theorem of multiple dimension functional format.This is the generalization from the one dimension monotone class theorem. It is also apowerful tool for the study of multi-dimension measurable functions and multi-dimensionstochastic process. We apply it to the study of VSI and improve the proof of manytheorems. Comparing to the original proofs, the new proofs are significantly simplifiedand easy to understand.
2. New properties and improvements on VSI. Combining the density processof local absolute continuous measure, properties of the martingale space H1 and semi-martingale space, and the closed image theorem of functional analysis (see the proofs ofTheorem 2.4.5 and Theorem 2.4.10), we obtain the general form of Girsanov Theorem forsemi-martingale vector stochastic integral(SMVSI) . This result is of particular value tothe theory of stochastic analysis and the application of stochastic integral. It also providesthe theoretical foundation for our study in mathematical finance.
3. The semi-martingale predictable representation theorem (Theorem3.4.5). This theorem is derived from the functional representation theorem in martin-gale space H1 , the expansion theorem in functional analysis, and the Girsanov Theoremfor SMVSI (Theorem 2.4.10). Many techniques in stochastic analysis are used to obtainthis result, which also has high theoretical and practical value. Using this result, we areable to extend the characteristics of the market completeness.
4. Summary of important existing results. We give the relationships of VSI un-der several di?erent meanings and list the most commonly used special cases. More specif- ically, this includes equivalent martingale measure, equivalent local martingale measure,local absolutely continuous local martingale measure, equivalent martingale transformmeasure, equivalent local martingale transform measure, and local absolutely continuouslocal martingale transform measure. Two deeper theorems (Theorem 3.2.1 and Theorem3.2.7) are obtained from our Girsanov Theorem for SMVSI.
5. Proof of theorem on how to derive the d+1 dimension self-financingstrategy from a d-dimension predictable process (Theorem 3.1.5). After a carefulstudy of relationships between the original market and the deflated market, and theircorresponding strategies (self-financing strategy and admissible strategy, respectively),we discuss, from both mathematics and economics point of view, the first fundamentaltheorem of asset pricing in different forms and their relationship.
引文
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