Some spectral ideas applied to finance and to self-similar and long-range dependent processes

This dissertation consists of four parts. The aim of the first part is to present original transformations 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 enables 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 applying 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 subordinated 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 (presence 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 generalized 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 spectrum 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 relations, 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 ∈ (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 Lamperti mapping to this Bernstein function. Resorting to spectral theoretical arguments, we investigate the generalized Cauchy problems, defined with these self-similar multiplicative convolution operators. In particular, we provide both a stochastic representation, expressed 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