Uniform tail asymptotics for the stochastic present value of aggregate claims in the renewal risk model

Consider an insurer who is allowed to make risk-free and risky investments. The price process of the investment portfolio is described as a geometric Lévy process. We study the tail probability of the stochastic present value of future aggregate claims. When the claim-size distribution is of Pareto type, we obtain a simple asymptotic formula which holds uniformly for all time horizons. The same asymptotic formula holds for the finite-time and infinite-time ruin probabilities. Restricting our attention to the so-called constant investment strategy, we show how the insurer adjusts his investment portfolio to maximize the expected terminal wealth subject to a constraint on the ruin probability.     

On an application of Hall's representatives theorem to a finite geometry problem

It is well-known thatn points not belonging to a hyperplane determine at leastn hyperplanes. The possible configurations of hyperplanes in the case when the number of hyperplanes is equal ton are known, too. In this paper we obtain these results by means of Hall's representatives theorem. The setting is that of a finite geometry.     

On a general principle in geometry that leads to functional equations

Summary Suppose that an invariant (or an invariant notion) of some geometry is given, like the distance between two points, the cross ratio of four points, the tangential distance between two spheres (or like the notion of orthogonality, of order, of a circle). One may ask what are the functions preserving (or preserving partially) that invariant (invariant notion). Originating from this principle some functional equation problems are formulated, namely the functional equations of distance, of area, of angle preservance.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

Der Richtungsabstand

In the study of chemical structural phenomena, the idea of mixedness appears to provide most valuable information if this notion is understood as a quantity that counts for a natural distinction between more or less mixed situations. The search for such a concept was initiated by the need of a corresponding valuation of chemical molecules that differ in the type-composition of a system of varying molecular parts at given molecular skeleton sites. In other words, an order relation for the partitions of a finite set was sought that explains the extent of mixing in a canonical way. This and related questions led to the concepts of themixing character andmixing distance. Success in applying these concepts to further chemical and physical problems, to graph theory, to representation theory of the symmetric group, and to probability theory confirmed the hope that there is a common background in some basic mathematics that allows a systematic treatment.The expected concept summarizing the above-mentioned experience is called thedirection distance and the mathematics concerned is linear geometry with a normspecific metric or structural analysis of normed vector spaces, respectively. Direction distance is defined as a map that represents the total metric information on any pair of directions (= pair of half-lines with a common vertex or a corresponding figure in normed vector spaces). Generally, that metrical figure changes when the half-lines are interchanged. As a consequence thereof, Hilbert's congruence axioms do not permit a metric criterion for the congruence of angles except in particular cases. The metric figures of direction pairs, however, may be classified according to metric congruence, and the normspecific metric induces an order in the set of congruence classes. This order, as a rule, is partial; it proves to be total if and only if the vector spaces are (pre-) Hilbert spaces (Lemma 8). A thorough comparison of the direction distance with the conventional distance deepens the understanding of the novel concept and justifies the terminology. The results are summarized in a number of lemmata. Furthermore, so-calledd-complete systems of order-homomorphic functional (so-calledd-functionals) establish an alternative formulation of the direction distance order. If and only if the order is total,d-complete systems can be represented by singled-functionals. Consequently, the case that normed vector spaces are (pre-) Hilbert spaces is pinpointed by the fact that the negative scalar product is already ad-complete system. These particular circumstances allow a metric congruence relation for angles.Another family of normed vector spaces is traced out by the conditions under which the direction distance takes the part of the mixing distance. Roughly speaking, a subset of vectors may be viewed as representing mixtures if it has two properties. First, with any two vectors of this subset all positive linear combinations are vectors of it as well. Second, the length of these vectors is an additive property. Correspondingly, the definition of the mentioned family, the family of so-calledmc-spaces, is based on the concepts of ameasure cone (Def. 5 and Def. 5) and an associated class ofmc- (= measure cone)norms being responsible for length additivity ofpositive vectors (= vectors of the measure cone) (Def. 6). Such norms provide congruence classes for positive vectors and positive direction pairs marked by the propertieslength andmixing distance, respectively. These congruence classes do not depend on the choice of the particularmc-norm within the class associated with a given measure cone, however, the mixing distance does. The consistency of the stipulated mathematical instrumentarium becomes apparent with Theorem 1 stating: The mixing distanceorder doesnot depend on the choice of a particular norm within the measure cone specific class; this order, together with the stipulated length of positive vectors, are properties necessary and sufficient for fixing the measure cone specific class ofmc-norms.Decreasing (or constant) mixing distance was found to describe a characteristic change in the relation between two probability distributions on a given set of classical events, a change in fact necessary and sufficient for the existence of alinear stochastic operator that maps a given pair of distributions into another given pair. This physically notable statement was originally proved for the space ofL ¹-functions on a compact -interval, it was expected to keep its validity for probability distributions in the range of classical physics and, as a consequence of that, for measures of any type. Theorem 2 presents the said statement in terms ofmc-endomorphisms ofmc-spaces; after an extension of the original proof to a more general family ofL ¹-spaces another method presented in a separate paper confirms Theorem 2 for bounded additive set functions and, accordingly, secures the expected range of validity. The discussion below is without reference to the validity range and primarily devoted to geometrical consequences without detailed speculations about physical applications.A few remarks on applications, however, illustrate the physical relevance of the mixing distance and its specialization, theq-character, in the particular context of Theorem 2. With reference to measure cones with such physical interpretations as statistical systems,mc-endomorphisms effect changes that can be described by linear stochastic operators and result physically either from an approach to some equilibrium state or from an adoption to a time-dependent influence on the system from outside. Theorem 2 provides a necessary and sufficient criterion for such changes. The discussion may concern phenomena of irreversible thermodynamics as well as evolving systems under the influence of a surrounding world summarized asorganization phenomena. Entropies and relative entropies of the Renyi-type ared-functionais which do not establishd-complete systems. The validity of Theorem 2 does not encompass the nonclassical case; the reason for it is of high physical interest. The full range of validity and its connection with symmetry arguments seems a promising mathematical problem in the sense of Klein'sErlanger Programm. From the point of mathematical history, the Hardy-Littlewood-Polya theorem should be quoted as a very special case of Theorem 2.

     

Über die abstandsverträglichen Abbildungen auf dem Kreis und auf der reellen Geraden

Zusammenfassung. Eine Abbildung zwischen metrischen R?umen hei?t abstandsvertr?glich, wenn der Abstand der Bilder zweier Punkte nur vom Abstand der Punkte selbst abh?ngt. Wir zeigen, dass eine Abbildung genau dann abstandsvertr?glich ist, wenn der Cauchyschen Funktionalgleichung genügt, also ein Endomorphismus der Gruppe ist. Ein entsprechendes Resultat gilt auch für die abstandsvertr?glichen Abbildungen des Kreises (mit der Multiplikation komplexer Zahlen als Gruppenverknüpfung). Damit kann man sowohl alle messbaren abstandsvertr?glichen Abbildungen von bzw. in sich angeben, als auch einen Nachweis für die Existenz nichtmessbarer abstandsvertr?glicher Abbildungen auf und erbringen. Eingegangen am 20. Juni 2001 / Angenommen am 13. September 2001     

Transition probabilities of Lévy‐type processes: Parametrix construction

We present an existence result for Lévy‐type processes which requires only weak regularity assumptions on the symbol with respect to the space variable x. Applications range from existence and uniqueness results for Lévy‐driven SDEs with Hölder continuous coefficients to existence results for stable‐like processes and Lévy‐type processes with symbols of variable order. Moreover, we obtain heat kernel estimates for a class of Lévy and Lévy‐type processes. The paper includes an extensive list of Lévy(‐type) processes satisfying the assumptions of our results.     

Stitching pairs of Lévy processes into harnesses

We consider natural exponential families of Lévy processes with randomized parameter. Such processes are Markov, and under suitable assumptions, pairs of such processes with shared randomization can be “stitched together” into a single harness. The stitching consists of deterministic reparametrization of the time for both processes, so that they run on adjacent time intervals, and of the choice of the appropriate law at the boundary.     

Stochastic integration for Lévy processes with values in Banach spaces

A stochastic integral of Banach space valued deterministic functions with respect to Banach space valued Lévy processes is defined. There are no conditions on the Banach spaces or on the Lévy processes. The integral is defined analogously to the Pettis integral. The integrability of a function is characterized by means of a radonifying property of an integral operator associated with the integrand. The integral is used to prove a Lévy–Itô decomposition for Banach space valued Lévy processes and to study existence and uniqueness of solutions of stochastic Cauchy problems driven by Lévy processes.     

On the Hyers–Ulam stabilities of functional equations on ‐Banach spaces

The purpose of this article is to generalize the theory of stability of functional equations to the case of n‐Banach spaces. In this article, we prove the generalized Hyers–Ulam stabilities of the Cauchy functional equations, Jensen functional equations and quadratic functional equations on n‐Banach spaces.