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Spectral Expansions and Excursion Theory for Non-Self-Adjoint Markov Semigroups with Applications in Mathematical Finance

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Spectral Expansions and Excursion Theory for Non-Self-Adjoint Markov Semigroups with Applications in Mathematical Finance SPECTRAL EXPANSIONS AND EXCURSION THEORY FOR NON-SELF-ADJOINT MARKOV SEMIGROUPS WITH APPLICATIONS IN MATHEMATICAL FINANCE A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Yi Xuan Zhao December 2017 c 2017 Yi Xuan Zhao ALL RIGHTS RESERVED SPECTRAL EXPANSIONS AND EXCURSION THEORY FOR NON-SELF-ADJOINT MARKOV SEMIGROUPS WITH APPLICATIONS IN MATHEMATICAL FINANCE Yi Xuan Zhao, Ph.D. Cornell University 2017 This dissertation consists of three parts. In the first part, we establish a spectral 2 + theory in the Hilbert space L (R ) of the C0-semigroup P and its adjoint Pb hav- ing as generator, respectively, the Caputo and the right-sided Riemann-Liouville fractional derivatives of index 1 < α < 2. These linear operators, which are non- local and non-self-adjoint, appear in many recent studies in applied mathemat- ics and also arise as the infinitesimal generators of some substantial processes such as the reflected spectrally negative α-stable process. We establish an inter- twining relationship between these semigroups and the semigroup of a Bessel type process which is self-adjoint. Relying on this commutation identity, we characterize the spectrum and the (weak) eigenfunctions and provide the spec- tral expansions of these semigroups on (at least) a dense subset of L2(R+). We also obtain an integral representation of their transition kernels that enables to derive regularity properties. Inspired by this development, we further exploit, in the second part of this dissertation, the concept of intertwining between general Markov semigroups. More specifically, we start by showing that the intertwining relationship be- tween two minimal Markov semigroups acting on Hilbert spaces implies that any recurrent extensions, in the sense of Ito,ˆ of these semigroups satisfy the same intertwining identity. Under mild additional assumptions on the inter- twining operator, we prove that the converse also holds. This connection en- ables us to give an interesting probabilistic interpretation of intertwining rela- tionships between Markov semigroups via excursion theory: two such recurrent extensions that intertwine share, under an appropriate normalization, the same local time at the boundary point. Moreover, in the case when one of the (non- self-adjoint) semigroup intertwines with the one of a quasi-diffusion, we obtain an extension of Krein’s theory of strings by showing that its densely defined spectral measure is absolutely continuous with respect to the measure appear- ing in the Stieltjes representation of the Laplace exponent of the inverse local time. Finally, we illustrate our results with the class of positive self-similar Markov semigroups and also the reflected generalized Laguerre semigroups. For the latter, we obtain their spectral decomposition and provide, under some conditions, a perturbed spectral gap estimate for its convergence to equilibrium. The third part of this dissertation is devoted to the applications of some of these theoretical results to some substantial problems arising in financial math- ematics. Keeping in mind the fundamental theorem of asset pricing, we suggest several transformations on a tractable and flexible Markov process (or equiva- lently, its respective semigroup) in order that the discounted transformed pro- cess becomes a (local) martingale while still keeping its tractability. In partic- ular, we suggest using an intertwining approach and/or Bochner’s subordina- tion (random time-change via a subordinator) to achieve this goal. Moreover, in order to illustrate our approach, we discuss in details several examples that include the class of Levy,´ self-similar and generalized CIR processes that reveal the usefulness of our result. Furthermore, we provide for the non-self-adjoint pricing semigroups associated to the latter family of processes a spectral expan- sions on which we carry out some numerical analysis. BIOGRAPHICAL SKETCH Yi Xuan Zhao was born and raised in Dalian, a modern coastal city in northeast China which serves as an important naval base and logistics centre. He attended University of Waterloo studying mathematical finance and operations research, after which he continued to pursue a doctoral degree in the ORIE department at Cornell University. During his five years spent at Cornell, Yi Xuan worked with Professor Pierre Patie on properties of some non-self-adjoint Markov semigroups, mainly fo- cusing on their spectral properties, excursion theory and their applications in mathematical finance. Besides his academic work, he enjoys traveling around the world as much as spending time with his friends in cinema, theatres and restaurants. After his graduation, Yi Xuan will join Credit Suisse in New York City as a quantitative strategist. iii This document is dedicated to all Cornell graduate students. iv ACKNOWLEDGEMENTS My deepest gratitude goes to my advisor, Pierre Patie, for his guidance and support over the five years in Cornell. He motivates me not only in academic studies, but also in all aspects in life. His immense knowledge and great pa- tience have deeply impressed me, and his rigorous attitude towards work has influenced me so much. It would not be possible for me to achieve my academic performance without his help and encouragement. I would also like to thank my committee members, Robert Jarrow and Lau- rent Saloff-Coste, for offering their help in my research. Thank all the fac- ulty and staff at the Department of ORIE, especially Krishnamurthy Iyer, An- dreea Minca, Sidney Resnick, Gennady Samorodnitsky and Bruce Turnbull, with whom I have had the great pleasure to work as a teaching assistant. Special thanks to Peter Frazier, who offered me great help in my memorable experience working as the instructor of the Monte Carlo simulation course. Next, I thank all my Ph.D. fellow and friends at Cornell. Thank you my office mates Jiayang Gao, Nanjing Jian, Jiayi Guo and Eric Ni, with whom I share so many stories in life and so much cherishable memories and pleasure. Also thank Weici Hu, Bangrui Chen, Tiandong Wang, Sijia Ma, Michael Choi and Amy Zhang, for all your company and help. Last but not the least, I am grateful to my parents Hongxing Zhao and Mei Han, for their uoconditional love and support. v TABLE OF CONTENTS Biographical Sketch . iii Dedication . iv Acknowledgements . .v Table of Contents . vi 1 Introduction 1 1.1 Notations and preliminaries . .7 2 Spectral decomposition of fractional operators and a reflected stable semigroup 10 2.1 Introduction . 10 2.2 Fractional operators and the reflected stable semigroup . 14 2.3 Intertwining relationship . 21 2.3.1 The Markov multiplicative operator Λα ........... 23 2.3.2 Proofs of Theorem 2.3.1(2) and (3) . 27 2.4 Eigenfunctions and upper frames . 29 2.5 Co-residual functions . 32 2.6 Spectral representation, heat kernel and smoothness properties . 36 2.6.1 Spectral expansions of P and Pb in Hilbert spaces and the heat kernel . 36 2.6.2 Regularity properties . 39 2.7 The α-Bessel semigroup and the operator Hα ............ 42 3 Intertwining, excursion theory and Krein theory of strings for non- self-adjoint Markov semigroups 45 3.1 Introduction . 45 3.1.1 Preliminaries . 49 3.2 Statements of main results . 53 3.2.1 Intertwining relations and inverse local time . 53 3.2.2 Excursion theory . 55 3.2.3 Krein’s spectral theory of strings . 56 3.3 Proof of Theorem 3.2.1 and its corollaries . 59 3.3.1 Proof of Theorem 3.2.1 . 59 3.3.2 Proof of corollaries . 67 3.4 Reflected self-similar and generalized Laguerre semigroups . 70 3.4.1 Proof of Theorem 3.4.1(1), (2) and (3) . 77 3.4.2 Proof of Theorem 3.4.1(4) . 83 3.4.3 Proof of Theorem 3.4.1(5) and spectral expansions . 86 vi 4 Risk-neutralization Techniques and Examples 101 4.1 Introduction . 101 4.1.1 Preliminaries . 104 4.2 Risk-neutralization transformations . 109 4.2.1 An intertwining approach . 110 4.3 Examples . 115 4.3.1 Levy´ processes . 115 4.3.2 Self-Similar Processes . 117 4.3.3 Generalized Cox-Ingersoll-Ross Models . 123 4.4 Pricing Derivatives . 130 Bibliography 134 vii CHAPTER 1 INTRODUCTION The spectral theory of linear operators has been an essential subject in func- tional analysis. First introduced by David Hilbert in his original formulation of Hilbert space theory, it has revealed its importance in various fields of mathe- matics, including differential equations, probability theory, harmonic and com- plex analysis, etc. Moreover, it is also widely used in a large number of dif- ferent application areas, such as superconductivity, fluid mechanics, quantum mechanics, kinetic theory, and more recently, financial mathematics. In functional analysis, the motivation for studying spectral theory is to un- derstand the structure of a linear operator. Such a structure often includes, but is not restricted to, classifying the operator by means of equivalent transforma- tions, reconstructing the operator in simple forms such as a direct sum or direct integral, identifying invariant subspaces, characterizing a basis for its domain or range, etc. For finite dimensional matrices, these problems can be solved by means of eigenvalues and their corresponding eigenvectors. While in the case of a infinite-dimensional functional operator, the idea of eigenvalues general- izes into the so-called spectrum, whose formal definition will be given later in the context. Moreover, spectral theory enables us to classify a bounded normal operator by means of unitary equivalence, or represent it as an integral with respect to projection-valued measures over its spectrum, i.e. the resolution of identity. Such results are analogous to the eigendecomposition for finite dimen- sional matrices. However, the classical spectral theory is mainly focused on normal operators in Hilbert spaces. For the class of non-self-adjoint (NSA) operators, on the con- 1 trary, there has been only limited results, due to their non-local and non-normal nature, which brings significant difficulty in studying their spectral properties.
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