SOME SPECTRAL IDEAS APPLIED TO FINANCE AND TO SELF-SIMILAR AND LONG-RANGE DEPENDENT PROCESSES 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 Anna Srapionyan May 2019 ⃝c 2019 Anna Srapionyan ALL RIGHTS RESERVED SOME SPECTRAL IDEAS APPLIED TO FINANCE AND TO SELF-SIMILAR AND LONG-RANGE DEPENDENT PROCESSES Anna Srapionyan, Ph.D. Cornell University 2019 This dissertation consists of four parts. The aim of the first part is to present original trans- formations on tractable Markov processes (or equivalently, on their semigroup) in order to make the discounted transformed process a martingale, while keeping its tractability. We refer to such procedures as risk-neutral pricing techniques. To achieve our goal, we resort to the concept of intertwining relationships between Markov semigroups that en- ables us, on the one hand to characterize a risk-neutral measure and on the other hand to preserve the tractability and flexibility of the models, two attractive features of models in mathematical finance. To illustrate the usefulness of our approach, we proceed by ap- plying this risk-neutral pricing techniques to some classes of Markov processes that have been advocated in the literature as substantial models. In the second part, we introduce spectral projections correlation functions of a stochastic process which are expressed in terms of the non-orthogonal projections into eigenspaces of the expectation operator of the process and its adjoint. We obtain closed- form expressions of these functions involving eigenvalues, the condition number and/or the angle between the projections, along with their large time asymptotic behavior for three important classes of processes: general Markov processes, Markov processes sub- ordinated in the sense of Bochner and non-Markovian processes which are obtained by time-changing a Markov process with an inverse of a subordinator. This enables us to provide a unified and original framework for designing statistical tests that investigate critical properties of a stochastic process, such as the path properties of the process (pres- ence of jumps), distance from symmetry (self-adjoint or non-self-adjoint) and short-to- long-range dependence. To illustrate the usefulness of our results, we apply them to gen- eralized Laguerre semigroups, which is a class of non-self-adjoint and non-local Markov semigroups, and also to their time-change by subordinators and their inverses. In the third part, we introduce and study non-local Jacobi operators, which generalize the classical (local) Jacobi operator on [0; 1]. We show that these operators extend to the generator of an ergodic Markov semigroup with an invariant probability measure β and study its spectral and convergence properties. In particular, we give a series expansion of the semigroup in terms of explicitly defined polynomials, which are counterparts of the classical Jacobi orthogonal polynomials, and give a complete characterization of the spec- trum of the non-self-adjoint generator and semigroup in L2(β). We show that the variance decay of the semigroup is hypocoercive with explicit constants which provides a natural generalization of the spectral gap estimate. After a random warm-up time the semigroup also decays exponentially in entropy and is both hypercontractive and ultracontractive. All of our proofs hinge on developing commutation identities, known as intertwining re- lations, between local and non-local Jacobi operators/semigroups, with the local Jacobi operator/semigroup serving as a reference object for transferring properties to the non- local ones. In the last part, by observing that the fractional Caputo derivative of order α 2 (0; 1) can be expressed in terms of a multiplicative convolution operator, we introduce and study a class of such operators which also have the same self-similarity property as the Caputo derivative. We proceed by identifying a subclass which is in bijection with the set of Bernstein functions and we provide several representations of their eigenfunctions, expressed in terms of the corresponding Bernstein function, that generalize the Mittag- Leffler function. Each eigenfunction turns out to be the Laplace transform of the right- inverse of a non-decreasing self-similar Markov process associated via the so-called Lam- perti mapping to this Bernstein function. Resorting to spectral theoretical arguments, we investigate the generalized Cauchy problems, defined with these self-similar multiplica- tive convolution operators. In particular, we provide both a stochastic representation, ex- pressed in terms of these inverse processes, and an explicit representation, given in terms of the generalized Mittag-Leffler functions, of the solution of these self-similar Cauchy problems. BIOGRAPHICAL SKETCH Anna Srapionyan was born and raised in Yerevan, the stunning capital city of the Repub- lic of Armenia which, being 2800 years old, showcases the beauty and historical values of Armenia. Anna received her Bachelor’s degree in Mathematics from Yerevan State Uni- versity, and her Master of Science in Financial Mathematics degree from The University of Chicago, after which she continued to pursue a doctoral degree at the Center for Applied Mathematics at Cornell University. During her five years at Cornell, Anna worked with Professor Pierre Patie on risk- neutral pricing techniques, spectral projections correlations structure for short-to-long range dependent processes, on non-local ergodic Jacobi semigroups, and self-similar Cauchy problems. Besides her academic work, she enjoys baking for her friends, hik- ing in the gorgeous gorges of Ithaca, reading and traveling. After her graduation, Anna will join Bank of America Merrill Lynch in New York City as a quantitative strategies associate. iii This document is dedicated to all my teachers and professors. iv ACKNOWLEDGEMENTS Completion of this doctoral dissertation was possible with the support of several people. First of all, I want to extend my special appreciation and thanks to my advisor, Pierre Patie, for his invaluable guidance, patience and consistent encouragement I received throughout the research work over the past five years at Cornell. His supervision and professional dedication has opened up new horizons of opportunities in front of me to enhance the quality of my research and motivated me to never give up. My academic achievements would not be possible without his tremendous mentorship and advice. I would also like to thank my committee members, Robert A. Jarrow, Andreea Minca and Gennady Samorodnitsky, for their great support, insightful discussions and invalu- able advice. Special thanks to the Engineering Learning Initiatives and to each one of my students who made the teaching experience one of the highlights of my years as a graduate student and who were the reason for teaching to become one of my passions. My time at Cornell was made enjoyable in large part due to the many friends and groups that became a part of my life. I would like to express my deepest gratitude to my Ph.D. colleagues and friends at Cornell, to the wonderful Armenian community of Ithaca and to my dear friend Xinrong Zhu for their company, for the laughs and all the unforgettable moments. Finally yet most importantly, I am grateful to my parents William Srapionyan and Aghavni Yeritsyan, as well as my younger brothers Nerses Srapionyan and Hamlet Sra- pionyan, for loving me unconditionally, for trusting me and for the wings they have given me. v CONTENTS Biographical Sketch .................................... iii Dedication ......................................... iv Acknowledgements .................................... v Contents .......................................... vi 1 Introduction 1 2 Risk-neutral pricing techniques and examples 10 2.1 Introduction ..................................... 10 2.1.1 Preliminaries ................................ 12 2.2 Risk-neutral pricing transformations ....................... 16 2.3 Examples ....................................... 21 2.3.1 Exponential of Lévy processes ...................... 22 2.3.2 Positive self-similar Markov processes .................. 25 2.3.3 Generalized Cox-Ingersoll-Ross models ................. 30 2.4 Option pricing in the jump CIR models ..................... 34 2.5 Proofs of the main results ............................. 40 2.5.1 Proof of Theorem 2.2.1 and Proposition 2.2.1 .............. 40 2.5.2 Extended generators and resolvents ................... 41 2.5.3 Proof of Proposition 2.2.3 ......................... 45 3 Spectral projections correlation structure for short-to-long range dependent pro- cesses 47 3.1 Introduction ..................................... 47 3.1.1 Preliminaries ................................ 49 3.1.2 Covariance and correlation functions .................. 55 3.2 Main results ..................................... 58 3.2.1 Interpretation of the (biorthogonal) spectral projections correlation functions for statistical properties .................... 64 3.3 Examples ....................................... 67 3.3.1 A short review of the generalized Laguerre semigroups ........ 69 3.3.2 The self-adjoint diffusion case ....................... 71 3.3.3 Small perturbation of the Laguerre semigroup. ............. 73 3.3.4 The Gauss-Laguerre semigroup. ..................... 74 3.4 Proofs of the main results ............................. 76 3.4.1 Proof of Theorem 3.2.1 ..........................