Vector Stochastic Integrals And The Fundamental Theorems Of Asset Pricing

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.