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Radically Elementary Stochastic Summation with Applications to Finance

Title: Radically Elementary Stochastic Summation with Applications to Finance.
Name(s): Zhu, Ming, author
Nichols, Warren D., professor co-directing dissertation
Kim, Kyounghee, professor co-directing dissertation
Huffer, Fred W., university representative
Ewald, Brian, committee member
Kercheval, Alec N., committee member
Okten, Giray, committee member
Department of Mathematics, degree granting department
Florida State University, degree granting institution
Type of Resource: text
Genre: Text
Issuance: monographic
Date Issued: 2014
Publisher: Florida State University
Place of Publication: Tallahassee, Florida
Physical Form: computer
online resource
Extent: 1 online resource
Language(s): English
Abstract/Description: This dissertation develops a nonstandard approach to probability, stochastic calculus and financial modeling, within the framework of the Radically Elementary Probability Theory of Edward Nelson. The fundamental objects of investigation are stochastic sums with respect to a martingale, defined on a finite probability space and indexed by a finite set. We study the external (nonstandard) properties of these sums, such as almost sure continuity of trajectories, the Lp property, and the Lindeberg condition; we also study external properties of related processes, such as quadratic variation and proper time. Using the tools so developed, we obtain an Itô-Doeblin formula for change of variable and a Girsanov theorem for change of measure in a quite general setting. We also obtain results that will aid us in the comparison of certain of the processes we investigate to conventional ones. We illustrate the theory by using general techniques to build stock models driven by Wiener walks, Poisson walks and their combinations, and show in each case that when our parameter processes are constant we recover the prices for European calls of the corresponding models that use conventional stochastic calculus. Finally, we exhibit a model driven by a nonstandard Wiener process that produces different prices for European calls than are given by the conventional Black-Scholes model.
Identifier: FSU_migr_etd-9125 (IID)
Submitted Note: A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
Degree Awarded: Summer Semester, 2014.
Date of Defense: July 15, 2014.
Keywords: Financial Modeling, Nonstandard Analysis, Option Pricing, Probability, Stochastic Process, Stochastic Summation
Bibliography Note: Includes bibliographical references.
Advisory Committee: Warren D. Nichols, Professor Co-Directing Dissertation; Kyounghee Kim, Professor Co-Directing Dissertation; Fred W. Huffer, University Representative; Brian Ewald, Committee Member; Alec N. Kercheval, Committee Member; Giray Okten, Committee Member.
Subject(s): Mathematics
Persistent Link to This Record:
Owner Institution: FSU

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Zhu, M. (2014). Radically Elementary Stochastic Summation with Applications to Finance. Retrieved from