On Lorentz geometry in algebras of generalized functions

We introduce a concept of causality in the framework of generalized pseudo-Riemannian Geometry in the sense of J.F. Colombeau and establish the inverse Cauchy-Schwarz inequality in this context. As an application, we prove a dominant energy condition for some energy tensors as put forward in Hawking and Ellis's book "The large scale structure of space-time". Our work is based on a new characterization of free elements in finite dimensional modules over the ring of generalized numbers.Comment: 26 pages, reorganized, updated reference

Spherical completeness of the non-archimedean ring of Colombeau generalized numbers

We show spherical completeness of the ring of Colombeau generalized real (or complex) numbers endowed with the sharp norm. As an application, we establish a Hahn Banach extension theorem for ultra pseudo normed modules (over the ring of generalized numbers) of generalized functions in the sense of Colombeau.Comment: 11 pages, parts rewritten, notation improve

How singular are moment generating functions?

This short note concerns the possible singular behaviour of moment generating functions of finite measures at the boundary of their domain of existence. We look closer at Example 7.3 in O. Barndorff-Nielsen's book "Information and Exponential Families in Statistical Theory (1978)" and elaborate on the type of exhibited singularity. Finally, another regularity problem is discussed and it is solved through tensorizing two Barndorff- Nielsen's distributions

Rational Decompositions of p-adic meromorphic functions

Let K be a non archimedean algebraically closed field of characteristic pi complete for its ultrametric absolute value. In a recent paper by Escassut and Yang, polynomial decompositions P(f)=Q(g) for meromorphic functions f, g on K (resp. in a disk) have been considered, and for a class of polynomials P, Q, estimates for the Nevanlinna function T(r,f) have been derived. In the present paper we consider as a generalization rational decompositions of meromorphic functions. In the case, where f, g are analytic functions, the Second Nevanlinna Theorem yields an analogue result as in the mentioned paper. However, if they are meromorphic, non trivial estimates for T(r,f) are more sophisticated.Comment: 13 page

Affine processes on positive semidefinite d x d matrices have jumps of finite variation in dimension d > 1

The theory of affine processes on the space of positive semidefinite d x d matrices has been established in a joint work with Cuchiero, Filipovi\'c and Teichmann (2011). We confirm the conjecture stated therein that in dimension d greater than 1 this process class does not exhibit jumps of infinite total variation. This constitutes a geometric phenomenon which is in contrast to the situation on the positive real line (Kawazu and Watanabe, 1974). As an application we prove that the exponentially affine property of the Laplace transform carries over to the Fourier-Laplace transform if the diffusion coefficient is zero or invertible.Comment: version to appear in Stochastic Processes and Their Application

Reforming the Wishart characteristic function

The literature presents the characteristic function of the Wishart distribution on m times m matrices as an inverse power of the determinant of the Fourier variable, the exponent being the positive, real shape parameter. I demonstrate that only for two times two matrices, this expression is unambiguous -- in this case the complex range of the determinant excludes the negative real line. When m greater or equals 3 the range of the determinant contains closed lines around the origin, hence a single branch of the complex logarithm does not suffice to define the determinant's power. To resolve this issue, I give the correct analytic extension of the Laplace transform, by exploiting the Fourier-Laplace transform of a Wishart process

Rational decompositions of complex meromorphic functions

Let h be a complex meromorphic function decomposed in two different ways P(f) and Q(g), where f, g are meromorphic functions and P, Q are rational functions. We follow an approach due to C.-C. Yang, P. Li and K. H. Ha who handle similar decompositions, however, with P, Q polynomials, by applying the Second Nevanlinna Theorem. We establish relations between the number k of distinct zeros of P', the degrees of P and Q, and (unless f, g are analytic functions) the number of distinct zeros of the denominators of P and Q.Comment: 5 page

A Reduced Basis Method for Parabolic Partial Differential Equations with Parameter Functions and Application to Option Pricing

We consider the Heston model as an example of a parameterized parabolic partial differential equation. A space-time variational formulation is derived that allows for parameters in the coefficients (for calibration) as well as choosing the initial condition (for option pricing) as a parameter function. A corresponding discretization in space and time amd initial condition is introduced and shown to be stable. Finally, a Reduced Basis Method (RBM) is introduced that is able to use parameter functions also for the initial condition. Corresponding numerical results are shown

Affine Processes

We put forward a complete theory on moment explosion for fairly general state-spaces. This includes a characterization of the validity of the affine transform formula in terms of minimal solutions of a system of generalized Riccati differential equations. Also, we characterize the class of positive semidefinite processes, and provide existence of weak and strong solutions for Wishart SDEs. As an application, we answer a conjecture of M.L. Eaton on the maximal parameter domain of non-central Wishart distributions. The last chapter of this thesis comprises three individual works on affine models, such as a characterization of the martingale property of exponentially affine processes, an investigation of the jump-behaviour of processes on positive semidefinite cones, and an existence result for transition densities of multivariate affine jump-diffusions and their approximation theory in weighted Hilbert spaces

On the existence of non-central Wishart distributions

This paper deals with the existence issue of non-central Wishart distributions which is a research topic initiated by Wishart (1928), and with important contributions by e.g., L\'evy (1937), Gindikin (1975), Shanbhag (1988), Peddada and Richards (1991). We present a new method involving the theory of affine Markov processes, which reveals joint necessary conditions on shape and non-centrality parameter. While Eaton's conjecture concerning the necessary range of the shape parameter is confirmed, we also observe that it is not sufficient anymore that it only belongs to the Gindikin ensemble, as is in the central case.Comment: This version contains an Appendix which explains the relation of my definition of non-central Wishart distributions to alternative ones from the standard literatur