Three expressions are provided for the first hitting time density of an Ornstein-Uhlenbeck process to reach a fixed level. The first hinges on an eigenvalue expansion involving zeros of the parabolic cylinder functions. The second is an integral representation involving some special functions whereas the third is given in terms of a functional of a 3-dimensional Bessel bridge. The expressions are used for approximating the density.

In this work we analyse the solution to the recurrence equationMΨ(z), MΨ(1) = 1, defined on a subset of the imaginary line and where −Ψ runs through the set of all continuous negative definite functions. Using the analytic Wiener-Hopf method we furnish the solution to this equation as a product of functions that extend the classical gamma function. These latter functions, being in bijection with the class of Bernstein functions, are called Bernsteingamma functions. Using their Weierstrass product representation we establish universal Stirling type asymptotic which is explicit in terms of the constituting Bernstein function. This allows the thorough understanding of the decay of |MΨ(z)| along imaginary lines and an access to quantities important for many theoretical and applied studies in probability and analysis.This functional equation appears as a central object in several recent studies ranging from analysis and spectral theory to probability theory. In this paper, as an application of the results above, we investigate from a global perspective the exponential functionals of Lévy processes whose Mellin transform satisfies the equation above. Although these variables have been intensively studied, our new approach based on a combination of probabilistic and analytical techniques enables us to derive comprehensive properties and strengthen several results on the law of these random variables for some classes of Lévy processes that could be found in the literature. These encompass smoothness for its density, regularity and analytical properties, large and small asymptotic behaviour, including asymptotic expansions, bounds, and Mellin-Barnes representations of its successive derivatives. We also furnish a thorough study of the weak convergence of exponential functionals on a finite time horizon when the latter expands to infinity. As a result of new Wiener-Hopf and infinite product factorizations of the law of the exponential functional we deliver important intertwining relation between members of the class of positive self-similar semigroups. Some of the results presented in this paper have been announced in the note [60].

We provide the spectral expansion in a weighted Hilbert space of a substantial class of invariant non-self-adjoint and non-local Markov operators which appear in limit theorems for positive-valued Markov processes. We show that this class is in bijection with a subset of negative definite functions and we name it the class of generalized Laguerre semigroups. Our approach, which goes beyond the framework of perturbation theory, is based on an in-depth and original analysis of an intertwining relation that we establish between this class and a self-adjoint Markov semigroup, whose spectral expansion is expressed in terms of the classical Laguerre polynomials. As a by-product, we derive smoothness properties for the solution to the associated Cauchy problem as well as for the heat kernel. Our methodology also reveals a variety of possible decays, including the hypocoercivity type phenomena, for the speed of convergence to equilibrium for this class and enables us to provide an interpretation of these in terms of the rate of growth of the weighted Hilbert space norms of the spectral projections. Depending on the analytic properties of the aforementioned negative definite functions, we are led to implement several strategies, which require new developments in a variety of contexts, to derive precise upper bounds for these norms.

The question of the measurement of strategic long-term financial risks is of considerable importance. Existing modelling instruments allow for a good measurement of market risks of trading books over relatively small time intervals. However, these approaches may have severe deficiencies if they are routinely applied to longer time periods. In this paper we give an overview on methodologies that can be used to model the evolution of risk factors over a one-year horizon. Different models are tested on financial time series data by performing backtesting on their expected shortfall predictions.

We first introduce and derive some basic properties of a two-parameters (α, γ) family of one-sided Lévy processes, with 1 < α < 2 and γ > −α. Their Laplace exponents are given in terms of the Pochhammer symbol as followsstands for the Pochhammer symbol and Γ for the gamma function. These are a generalization of the Brownian motion, since in the limit case α → 2, we end up to Brownian motion with drift γ + 1 2 . Then, we proceed by computing the density of the law of the exponential functional associated to some elements of this family (and their dual) and some transformations of these elements. More specifically, we shall consider the Lévy processes which admit the following Laplace exponent, for any δ > α−1 α ,These densities are expressed in terms of the Wright hypergeometric function 1Ψ1. By means of probabilistic arguments, we derive some interesting properties enjoyed by this function. On the way we also characterize explicitly the semi-group of the family of selfsimilar positive Markov processes associated, via the Lamperti mapping, to the family of Lévy processes with Laplace exponent ψ (0,δ) .