A Weak Existence Result With Application To The Financial Engineer's Calibration Problem

Abstract A Weak Existence Result with Application to the Financial Engineer’s Calibration Problem Gerard Brunick Advisor: Steven E. Shreve Given an initial Itˆo process, Krylov and Gy¨ongy have shown that it is often possible to construct a diffusion process with the same one-dimensional marginal distributions. As the one-dimensional marginal distributions of a price process under a pricing measure essentially determine the prices of European options written on that price process, this result has found wide application in Mathematical Finance. In this dissertation, we extend the result of Krylov and Gy¨ongy in two directions: We relax the technical conditions which must be imposed on the initial Itˆo process. And we clarify the relationship between the stochastic differential equation that is solved by the mimicking process and the properties of the initial process that are preserved. A Weak Existence Result with Application to the Financial Engineer’s Calibration Problem Gerard Brunick Advisor: Steven E. Shreve Defense Date: July 29th , 2008 A dissertation in the Department of Mathematical Sciences submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at Carnegie Mellon University. Copyright © 2008 by Gerard Brunick All rights reserved. Abstract A Weak Existence Result with Application to the Financial Engineer’s Calibration Problem Gerard Brunick Advisor: Steven E. Shreve Given an initial Itˆo process, Krylov and Gy¨ongy have shown that it is often possible to construct a diffusion process with the same one-dimensional marginal distributions. As the one-dimensional marginal distributions of a price process under a pricing measure essentially determine the prices of European options written on that price process, this result has found wide application in Mathematical Finance. In this dissertation, we extend the result of Krylov and Gy¨ongy in two directions: We relax the technical conditions which must be imposed on the initial Itˆo process. And we clarify the relationship between the stochastic differential equation that is solved by the mimicking process and the properties of the initial process that are preserved. i Acknowledgments I would like to express my gratitude to my adviser, Steven Shreve, for his guidance and support as I worked on this dissertation. His comments and insight have been invaluable. I would also like to thank Dmitry Kramkov and Kasper Larson for many useful conversations on a wide range of topics. Finally, I would like acknowledge Peter Carr who made me aware of the previous work of Krylov and Gy¨ongy, and Silviu Predoiu who produced a very nice counterexample that allowed me to abandon a fallacious conjecture. I would also like to take this opportunity to thank my family for their love and encouragement during my time at Carnegie Mellon University. In particular, I would never have had this opportunity without my parents’ constant love, patience, and support. Finally, I would like to express my gratitude to Jessica whose love and kindness have been a constant source of inspiration. During my time at Carnegie Mellon University I was supported by an NSF VIGRE fellowship and grant DMS-0404682. ii Contents 1 Introduction 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Definitions and Notation . . . . . . . . . . . . . . . . . . . . . 1 1 7 2 Statement of Results 12 2.1 Updating Functions . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3 Applications to Mixture Models . . . . . . . . . . . . . . . . . 30 3 A Cross Product Construction 33 3.1 The Binary Construction . . . . . . . . . . . . . . . . . . . . . 36 3.2 Properties Preserved by the Binary Construction. . . . . . . . 42 3.3