Small-time expansions for the transition distributions of Lévy processes

Let X=(X_t)_t≥0 be a Lévy process with absolutely continuous Lévy measure ν. Small-time expansions of arbitrary polynomial order in t are obtained for the tails , y>0, of the process, assuming smoothness conditions on the Lévy density away from the origin. By imposing additional regularity conditions on the transition density p_t of X_t, an explicit expression for the remainder of the approximation is also given. As a byproduct, polynomial expansions of order n in t are derived for the transition densities of the process. The conditions imposed on p_t require that, away from the origin, its derivatives remain uniformly bounded as t→0. Such conditions are then shown to be satisfied for symmetric stable Lévy processes as well as some tempered stable Lévy processes such as the CGMY one. The expansions seem to correct the asymptotics previously reported in the literature.     

Opérateurs elliptiques sur les variétés non compactes

Soient M une variété, V un ouvert de M, et P un opérateur différentiel elliptique du second ordre, à coefficients C^∞ et réels tel que P1 0. Soit A_V l'opérateur induit par P dans l'espace de Banach C₀(V) des fonctions continues sur V nulles au point à l'infini de V, muni de la norme du suprémum. On démontre que A_V engendre un semi-groupe fortement continu à contraction ssi il existe K compact de V, h fonction continue strictement positive dans VβK et nulle au point à l'infini de V telle que (1 − P) h soit la distribution associée à une fonction continue non négative dans VβK. On en déduit immédiatement un résultat bien connu: si M est une variété de Cartan-Hadamard, A_M engendre un semi-groupe fortement continu à contraction dans C₀(M).Let M be a manifold, V an open set of M, and P an elliptic differential operator of the second order, with real C^∞ coefficients and such that P1 0. Let A_V be the operator induced by P in the complex Banach space C₀(V) of all continuous functions vanishing at the point at infinity of V, endowed with the supremum norm. One proves that A_V generates a strongly continuous contraction semi-group iff there exists K compact of V, h continuous strictly positive in VβK and 0 at infinity of V such that (1 − P) h is the distribution associated to a nonnegative continuous function in VβK. One deduces immediately from that a well-known result: if M is a Cartan-Hadamard manifold, A_M generates a strongly continuous contraction semigroup in C₀(M).     

LOCAL TIME ANALYSIS OF ADDITIVE LEVY PROCESSES WITH DIFFERENT L(E)VY EXPONENTS

Let X1 XN be independent, classical Levy processes on R^d with Levy exponents ψ1,…, ψN, respectively. The corresponding additive Levy process is defined as the following N-parameter random field on R^d, X（t） △= X1（t1） ＋ ... ＋ XN（tN）, At∈N. Under mild regularity conditions on the ψi＇s, we derive estimate for the local and uniform moduli of continuity of local times of X = {X（t）; t ∈R^N}.     

Critères d'induction et de coinduction pour certains anneaux d'opérateurs différentiels

Soit une algèbre de Lie opérant par dérivations sur un anneau local commutatif noethérien (R, ,K=R/ ), et soit V l'anneau des opérateurs différentiels construits à partir de R et . Posons ( )={d /d( ) } et ₀ = ( )/ : ₀ est une algèbre de Lie qui opère sur K par dérivations et l'on peut construire un anneau d'opérateurs différentiels sur K à l'aide de ₀, noté V₀. Grâce au (V₀ − V)-bimodule V/ V on définit l'induction (resp. la coinduction) de V₀ à V par Ind^V_v0=−_v0V/ V (resp. Coind^V_v0=Hom_v0(V/ V,−)) et on donne un critère pour qu'un V-module soit induit (resp. coinduit) à partir de V₀. Ces résultats sont des analogues de ceux, établis par Mackey pour les groupes de Lie et par Blattner pour les algèbres de Lie, analogues, basés sur la notion de système d'imprimitivité.Let be a Lie algebra acting by derivations on a commutative noetherian local ring (R, ,K=R/ ) and let V be the ring of differential operators built on R and . Defining ( )={d /d( ) } et ₀ = ( )/ : ₀ is a Lie algebra which acts on K by derivations, and we can construct a differential operators ring on K with ₀, denoted by V₀. With the help of the (V₀ − V)-bimodule V/ V we define the induction (resp. coinduction) from V₀ to V by Ind^V_v0=−_v0V/ V and we give a criterion for a V-module to be induced (resp. coinduced) from V₀. These results are similar to those established by Mackey for Lie groups and Blattner for Lie algebras, which are based on the notion of the system of imprimitivity.     

q-Bessel, arbres, et chemins valués

We give interpretations for quotient J_ν+1/J_ν of q-Bessel functions. These q-analogs are related to generating function of weighted complete binary trees according to the number of leaves and to multichains on a partially ordered set, corresponding to weighted paths in the plane.Nous donnons des interprétations combinatoires du rapport J_ν+1/J_ν de q-fonctions de Bessel. Ces q-analogues énumèrent des classes d'arbres binaires complets valués suivant le nombre de feuilles et des multichaînes d'un ensemble partiellement ordonné, correspondant à des chemins valués dans le plan.     

A Functional LIL for <Emphasis Type="Italic">d</Emphasis>-Dimensional Stable Processes; Invariance for Lévy- and Other Weakly Convergent Processes

We establish a functional LIL for the maximal process M(t) :=sup _0≤s≤t ‖X(s)‖ of an ℝ ^d -valued α-stable Lévy process X, provided X(1) has density bounded away from zero over some neighborhood of the origin. We also provide a broad invariance result governing a class independent-increment processes related to the domain of attraction of X(1). This breadth is particularly notable for two types of processes captured: First, it not only describes any partial sum process built from iid summands in the domain of normal attraction of X(1), but also addresses those with arbitrary iid summands in the full domain of attraction (here we give a technical condition necessary and sufficient for the partial sum process to share the exact LIL we prove for X). Second, it reveals that any Lévy process L such that L(1) satisfies the technical condition just mentioned will also share the LIL of X. Supported in part by NSF Grant DMS 02-05034.     

Principe de Hartogs dans les variétés CR

Let M be a CR manifold. The main results of this paper are the following:

When M is real analytic, a semi-global Hartogs extension phenomenon occurs for real analytic CR functions if and only if M is nowhere strictly pseudoconvex and .

When M is a standard manifold, the Hartogs–Bochner extension phenomenon occurs for non-CR-confined domains if and only if M is nowhere strictly pseudoconvex and dim_CRM2.

If M is a smooth submanifold of foliated by complex curves, a semi-global Hartogs–Bochner extension phenomenon occurs for smooth non-CR-confined domains if and only if dim_CRM2.

If M is a real analytic nowhere strictly pseudoconvex manifold and if Ω is a sufficiently small domain in M, a hyperfunction which is real analytic in a neighborhood of bΩ and CR in a neighborhood of is in fact real analytic on Ω.

Uniformly Gâteaux smooth approximations on

Every Lipschitz mapping from c₀(Γ) into a Banach space Y can be uniformly approximated by Lipschitz mappings that are simultaneously uniformly Gâteaux smooth and C^∞-Fréchet smooth.     

A countable Fréchet–Urysohn space of uncountable character

We shall construct a countable Fréchet–Urysohn α₄ not α₃ space X such that all finite powers of X are Fréchet–Urysohn.     

Problèmes linéaires dans le Compendy de la praticque des nombres de Barthélemy de Romans et Mathieu Préhoude (1471): Une approche nouvelle basée sur des sources proches du Liber abbaci de Léonard de Pise

The Compendy de la praticque des nombres (1471) is one of a number of commercial arithmetics produced in southern France during the medieval period. Its interest and originality rest in its treatment of problem-solving. The author of the text limited his treatment to an in-depth analysis of only a few types of problems, not treating particular cases but rather emphasizing general methods. The sources from which he drew were very close to the Liber abbaci of Leonardo Fibonacci and many of them were new to the southern French arithmetic tradition. Thus, the Compendy sheds new light on the transmission of arithmetic thought into Europe. Copyright 2000 Academic Press.Le Compendy de la praticque des nombres (1471) est un traité qui appartient au groupe des arithmétiques commerciales du Sud de la France. L'intérêt et l'originalité du texte résident dans la partie consacrée à la résolution de problèmes. L'auteur sélectionne quelques types de problèmes seulement, auxquels il consacre une longue étude, délaissant les cas particuliers pour privilégier les méthodes. Ce faisant, il utilise de nouvelles sources, proches du “Liber abbaci” de Léonard de Pise, étrangères aux autres arithmétiques françaises de la même famille. Le Compendy nous apporte ainsi un éclairage nouveau sur la transmission de l'algorisme. Copyright 2000 Academic Press.MSC subject classifications: 01A35; 01A40.     

Les débuts de l'histoire des mathématiques sur les scènes internationales et le cas de l'entreprise encyclopédique de Felix Klein et Jules Molk

At the beginning of the 20th century, the history of mathematics first appeared on different international scenes as a result of different initiatives of mathematicians, historians, and philosophers. By studying the way in which historians of mathematics took advantage of these opportunities, we have sought the contours, the lines of force, and the equilibria of this process of the internationalization of the history of mathematics on an institutional as well as intellectual plane.This search was focused through the study of the German- and French-language editions of the Encyklopädie der mathematischen Wissenschaften, which is one of the most remarkable examples dating from this period of an international collaboration for mathematics and its history. Copyright 1999 Academic Press.Au début du vingtième siècle l'histoire des mathématiques fait ses débuts sur différentes scènes internationales à l'occasion de différentes initiatives de mathématiciens, d'historiens, de philosophes. En étudiant la façon dont les historiens des mathématiques se saisirent de ces opportunités, nous avons cherché à comprendre les contours, les lignes de force, les équilibres de ce processus d'internationalisation de l'histoire des mathématiques, tant sur le plan institutionnel qu'intellectuel.Cette enquête a été précisée par l'étude des éditions en langues allemande et française de l'Encyklopä-die der mathematischen Wissenschaften qui est un des exemples les plus marquant de cette période d'une collaboration internationale, tant pour les mathématiques que leur histoire. Copyright 1999 Academic Press.MSC subject classifications: 01A60, 01A74.     

Estimation de Yule–Walker d'un CAR(p) observé à temps discret

Let (X(lδ), l=0,n) be a discrete observation at mesh δ>0 of X, a CAR(p). Classical Yule–Walker estimation are biased and must be corrected. Resultant estimators converge if T=nδ→+∞, are asymptotically normal with rate , and efficient. The diffusion coefficient is also estimated, with rate .     

Enveloppes convexes des processus gaussiens

Let X={X(t), t[0,1]} be a process on [0,1] and V_X=Conv{(t,x)t[0,1], x=X(t)} be the convex hull of its path.The structure of the set ext(V_X) of extreme points of V_X is studied. For a Gaussian process X with stationary increments it is proved that:

• The set ext(V_X) is negligible if X is non-differentiable.

• If X is absolutely continuous process and its derivative X′ is continuous but non-differentiable, then ext(V_X) is also negligible and moreover it is a Cantor set.

It is proved also that these properties are stable under the transformations of the type Y(t)=f(X(t)), if f is a sufficiently smooth function.     

Automorphisms inc 0,l 1 andm

It is proved that ifX=c ₀ orm and ifY andZ are subspaces ofX withX/Y andX/Z non-reflexive, then any isomorphism ofY ontoZ has an extension to an automorphism ofX. A dual result is obtained forX=l ₁. This research was partially supported by NSF-GP-8964.     

On the ω-problem for some types of non-metrizable spaces

The ω-problem on a topological space X consists in finding out whether there exists a function whose oscillation is equal to a given upper semi-continuous (USC) function f:X→[0,∞] vanishing at isolated points of X. If such F exists, we call it an ω-primitive for f. Unlike the case of metrizable spaces, an ω-primitive need not exist if X is not metrizable. We study the ω-problem for f taking the value ∞ in the case of ordinal space, products of regular “constancy” spaces and the wedge sums of such spaces. Some open problems are formulated.     

Symbolical Algebra as a Foundation for Calculus: D. F. Gregory's Contribution

Duncan Farquharson Gregory (1813–1844) was a mathematician and founder of the Cambridge Mathematical Journal. His 1841 Examples of the Processes of the Differential and Integral Calculus was an extensive revision of Peacock's 1820 textbook of a similar title. Among the new material in Gregory's version is an exposition of symbolical algebra, prominently featuring the method of “separation of symbols.” We examine Gregory's career; the mathematicians he influenced; and Servois and Murphy, who influenced him. We consider Gregory's use of separation of symbols in the Examples and consider whether he believed these techniques to be capable of providing an adequate foundation for calculus. © 2002 Elsevier Science (USA).Le mathématicien Duncan Farquharson Gregory (1813–1844) fut également fondateur du Cambridge Mathematical Journal. Son ouvrage publié en 1841, Examples of the Processes of the Differential and Integral Calculus, constitue une révision substantielle du livre de Peacock au titre similaire. L'un des apports du livre de Gregory est un exposé de l'algèbre symbolique, et en particulier de la méthode de séparation des symboles. Dans cet article nous passons en revue la carrière de Gregory et nous parlons des mathématiciens qu'il a influencés, ainsi que de l'influence que Servois et Murphy ont eu sur ses travaux. Nous considérons son usage de la séparation des symboles dans ses Examples, et nous examinons si Gregory croyait que ses techniques pouvaient constituer un fondement pour le calcul différentiel. © 2002 Elsevier Science (USA).MSC 1991 subject classifications: 01A55, 01A70, 01A50, 26-03, 26A03, 34-03, 34A30.     

A self-normalized Erd s—Rényi type strong law of large numbers

The original Erd s—Rényi theorem states that max_0kn∑^k+[clogn]_i=k+1X_i/[clogn]→α(c),c>0, almost surely for i.i.d. random variables {X_n, n1} with mean zero and finite moment generating function in a neighbourhood of zero. The latter condition is also necessary for the Erd s—Rényi theorem, and the function α(c) uniquely determines the distribution function of X₁. We prove that if the normalizing constant [c log n] is replaced by the random variable ∑^k+[clogn]_i=k+1(X²_i+1), then a corresponding result remains true under assuming only the exist first moment, or that the underlying distribution is symmetric.     

An optimal dividends problem with transaction costs for spectrally negative Lévy processes

We consider an optimal dividends problem with transaction costs where the reserves are modeled by a spectrally negative Lévy process. We make the connection with the classical de Finetti problem and show in particular that when the Lévy measure has a log-convex density, then an optimal strategy is given by paying out a dividend in such a way that the reserves are reduced to a certain level c₁ whenever they are above another level c₂. Further we describe a method to numerically find the optimal values of c₁ and c₂.