Showing papers in "Mathematische Annalen in 1998"

The fundamental theorem of asset pricing for unbounded stochastic processes

Abstract: The Fundamental Theorem of Asset Pricing states - roughly speaking - that the absence of arbitrage possibilities for a stochastic process S is equivalent to the existence of an equivalent martingale measure for S. It turns out that it is quite hard to give precise and sharp versions of this theorem in proper generality, if one insists on modifying the concept of "no arbitrage" as little as possible. It was shown in [DS94] that for a locally bounded R^d-valued semi-martingale S the condition of No Free Lunch with Vanishing Risk is equivalent to the existence of an equivalent local martingale measure for the process S. It was asked whether the local boundedness assumption on S may be dropped. In the present paper we show that if we drop in this theorem the local boundedness assumption on S the theorem remains true if we replace the term equivalent local martingale measure by the term equivalent sigma-martingale measure. The concept of sigma-martingales was introduced by Chou and Emery - under the name of "semimartingales de la classe (Sigma_m)". We provide an example which shows that for the validity of the theorem in the non locally bounded case it is indeed necessary to pass to the concept of sigma-martingales. On the other hand, we also observe that for the applications in Mathematical Finance the notion of sigma-martingales provides a natural framework when working with non locally bounded processes S. The duality results which we obtained earlier are also extended to the non locally bounded case. As an application we characterize the hedgeable elements. (author's abstract)

Estimates on Green functions and Poisson kernels for symmetric stable processes

Abstract: One of the most basic and most important subfamily of Levy processes is symmetric stable processes. A symmetric α-stable process X on Rn is a Levy process whose transition density p(t , x − y) relative to the Lebesgue measure is uniquely determined by its Fourier transform ∫ Rn e ix ·ξp(t , x )dx = e−t|ξ| α . Here α must be in the interval (0, 2]. When α = 2, we get a Brownian motion running with a time clock twice as fast as the standard one. Brownian motion plays a central role in modern probability theory and has numerous important applications in other scientific areas as well as in many other branches of mathematics. Thus it has been intensively studied. In this paper, symmetric stable processes are referred to the case when 0 < α < 2, unless otherwise specified. In the last few years there has been an explosive growth in the study of physical and economic systems that can be successfully modeled with the use of stable processes. Stable processes are now widely used in physics, operations research, queuing theory, mathematical finance and risk estimation. In some physics literatures, symmetric α-stable processes are called Levy flights, and they have been applied to a wide range of very complex physics issues, such as turbulent diffusion, vortex dynamics, anomalous diffusion in rotating flows, and molecular spectral fluctuations. In mathematical finance, stable processes can be used to model stock returns in incomplete market. For these and more applications of stable processes, please see the interesting book [14] by Janicki and Weron and the references therein and the recent article [15] by Klafter, Shlesinger and Zuomofen. In order to make precise predictions about natural phenomena and to better cope with these

The algebraic structure of non-commutative analytic Toeplitz algebras

Abstract: The non-commutative analytic Toeplitz algebra is the wot– closed algebra generated by the left regular representation of the free semigroup on n generators. We develop a detailed picture of the algebraic structure of this algebra. In particular, we show that there is a canonical homomorphism of the automorphism group onto the group of conformal automorphisms of the complex n-ball. The k-dimensional representations form a generalized maximal ideal space with a canonical surjection onto the ball of k × kn matrices which is a homeomorphism over the open ball analogous to the fibration of the maximal ideal space of H∞ over the unit disk. In [6, 17, 18, 20], a good case is made that the appropriate analogue for the analytic Toeplitz algebra in n non-commuting variables is the wotclosed algebra generated by the left regular representation of the free semigroup on n generators. The papers cited obtain a compelling analogue of Beurling’s theorem and inner–outer factorization. In this paper, we add further evidence. The main result is a short exact sequence determined by a canonical homomorphism of the automorphism group onto this algebra onto the group of conformal automorphisms of the unit ball of Cn. The kernel is the subgroup of quasi-inner automorphisms, which are trivial modulo the wot-closed commutator ideal. Additional evidence of analytic properties comes from the structure of k-dimensional (completely contractive) representations, which have a structure very similar to the fibration of the maximal ideal space of H∞ over the unit disk. An important tool in our analysis is a detailed structure theory for wot-closed right ideals. Curiously, left ideals remain more obscure. The non-commutative analytic Toeplitz algebra Ln is determined by the left regular representation of the free semigroup Fn on n generators z1, . . . , zn which acts on `2(Fn) by λ(w)ξv = ξwv for v, w in Fn. In particular, the algebra Ln is the unital, wot-closed algebra generated by the isometries Li = λ(zi) for 1 ≤ i ≤ n. This algebra and its norm-closed version (the noncommutative disk algebra) were introduced by Popescu [19] in an abstract sense in connection with a non-commutative von Neumann inequality and 1991 Mathematics Subject Classification. 47D25. March 9, 1997; October 9, 1997 final draft. First author partially supported by an NSERC grant and a Killam Research Fellowship. Second author partially supported by an NSF grant.

A nonlocal anisotropic model for phase transitions

Abstract: where u is a scalar density function on a domain of R and takes values in [−1, 1], W is a positive double-well potential which vanishes at ±1, and J is a positive, possibly anisotropic, interaction potential which vanishes at infinity (see paragraph 1.2 for precise definitions). The scalar function u represents the macroscopic density profile of a system which has two equilibrium pure phases described by the profiles u ≡ +1 and u ≡ −1. The integral ∫ W (u) at the right side of (1.1) forces a minimizer of F to take values close to +1 and −1 (phase separation), while the double integral represents an interaction energy which penalizes the spatial inhomogeneity of the system (surface tension). In equilibrium Statistical Mechanics functionals of the form (1.1) arise as free energies of continuum limits of Ising spin systems on lattices; in this setting u plays the role of a macroscopic magnetization density and J is a ferromagnetic Kac potential (see for instance [2] and references therein). We underline the analogy with the more familiar gradient theory for phase transition proposed in [9], where the free energy of the system is of the form

A strong unique continuation theorem for parabolic equations

Abstract: The unique continuation is best understood for second order elliptic operators. The classical paper by Carleman [7] established the strong unique continuation theorem for second order elliptic operators which need not have analytic coefficients. This was remarkable since it indicates that even a non-analytic solution of an elliptic equation can behave somewhat in an “analytic” manner. The powerful technique he used, the so-called “Carleman weighted inequality” has played a central role in later developments. In 1950’s, Aronszajn [4] and Cordes [11] generalized his result to higher dimensions. In recent years this classical subject caught new attention from a great number of people. Much efforts have been made to relax the smoothness hypotheses on the coefficients (see [6, 17, 43, 1, 22]). A seminal paper by Jerison and Kenig [26] has shown the unique continuation property for operators of the form ∆ + c(x ) with c ∈ L /2 loc where N ≥ 3 is the space dimension. Further improvements have been carried out in considering other classes of coefficients (Fefferman-Phong class and Kato class) [42, 8, 12, 29], in extending the result to operators with first derivative terms and variable leading coefficients [46, 47, 51, 52], and etc. (See [25, 15, 16, 28, 5, 27] and the references therein.) For second order linear parabolic operators with time-independent coefficients, the strong unique continuation property along with detailed estimates on the Hausdorff measures of nodal sets was reduced in [32, 33] to the previously established elliptic counterparts. The reduction from time-independent parabolic

Computing a Selmer group of a Jacobian using functions on the curve

Abstract: In general, algorithms for computing the Selmer group of the Jacobian of a curve have relied on either homogeneous spaces or functions on the curve. We present a theoretical analysis of algorithms which use functions on the curve, and show how to exploit special properties of curves to generate new Selmer group computation algorithms. The success of such an algorithm will be based on two criteria that we discuss. To illustrate the types of properties which can be exploited, we develop a $(1-\zeta_{p})$-Selmer group computation algorithm for the Jacobian of a curve of the form $y^{p}=f(x)$ where $p$ is a prime not dividing the degree of $f$. We compute Mordell-Weil ranks of the Jacobians of three curves of this form. We also compute a 2-Selmer group for the Jacobian of a smooth plane quartic curve using bitangents of that curve, and use it to compute a Mordell-Weil rank.

Averages of norms and quasi-norms

Abstract: We compute the number of summands in q-averages of norms needed to approximate an Euclidean norm. It turns out that these numbers depend on the norm involved essentially only through the maximal ratio of the norm and the Euclidean norm. Particular attention is given to the case q = ∞ (in which the average is replaced with the maxima). This is closely connected with the behavior of certain families of projective caps on the sphere.

Geometry of growth: approximation theorems for \(L^2\) invariants

Abstract: In this paper we study the problem of approximation of the $L^2$-topological invariants by their finite dimensional analogues. We obtain generalizations of the theorem of Luck, dealing with towers of finitely sheeted normal coverings. We prove approximation theorems, establishing relations between the homological invariants, corresponding to infinite dimensional representations and sequences of finite dimensional representations, assuming that their normalized characters converge. Also, we find an approximation theorem for residually finite $p$-groups ($p$ is a prime), where we use the homology with coefficients in a finite field $\fp$. We view sequences of finite dimensional flat bundles of growing dimension as examples of growth processes. We study a von Neumann category with a Dixmier type trace, which allows to describe the asymptotic invariants of growth processes. We introduce a new invariant of torsion objects, the torsion dimension. We show that the torsion dimension appears in general as an additional correcting term in the approximation theorems; it vanishes under some arithmeticity assumptions. We also show that the torsion dimension allows to establish non-triviality of the Grothendieck group of torsion objects.

Sharp estimates and the Dirac operator

Abstract: This paper establishes and extends a conjecture posed by M. Gromov which states that every riemannian metric $g$ on $S^n$ that strictly dominates the standard metric $g_0$ must have somewhere scalar curvature strictly less than that of $g_0$ . More generally, if $M$ is any compact spin manifold of dimension $n$ which admits a distance decreasing map $f:M \rightarrow S^n$ of non-zero degree, then either there is a point $x \in M$ with normalized scalar curvature $\tilde{\kappa}(x)< 1$ , or $M$ is isometric to $S^n$ . The distance decreasing hypothesis can be replaced by the weaker assumption $f$ is contracting on $2$ -forms. In both cases, the results are sharp. An explicit counterexample is given to show that the result is no longer valid if one replaces 2-forms by $k$ -forms with $k \geq 3$ .

On the semi-simplicity of the $U_p$ -operator on modular forms

Abstract: For and positive integers, let denote the -vector space of cuspidal modular forms of level and weight . This vector space is equipped with the usual Hecke operators , . If we need to consider several levels or weights at the same time, we will denote this by , or ! . If " is a prime number dividing , our $# is also known under the name %&# . One of our main results can be stated very easily: if ('*) and ",+ does not divide , then the operator $# is semi-simple. We can prove the same result for weight /. , under the assumption that certain crystalline Frobenius elements are semi-simple. Milne has shown in [11, 0 2] that this semi-simplicity is implied by Tate’s conjecture claiming that for 1 projective and smooth over a finite field of characteristic " , and 23 54 , 687:9 =@? AB C1D FEHGJILKNMO equals the order of PQ C1R SB at 2 . Ulmer proved in [17] that $# is semi-simple, for 5'T. and ",U not dividing , under the assumption of the Birch-Swinnerton-Dyer conjecture for elliptic curves over function fields in characteristic " . His method is quite different from ours: assuming that $# is not semi-simple, he really shows that the Birch-Swinnerton-Dyer conjecture does not hold for an explicitly given elliptic curve. The structure of our proof is as follows. Using the theory of newforms, the problem is shown to be equivalent to the problem of showing that, for a normalized newform V of weight , prime-to-" level and character W , the polynomial X UZY\[ #]X<^\W8 _" `" !ba,c has no double root. This polynomial happens to be the characteristic polynomial of the Frobenius element at " in the two-dimensional Galois representations associated to V ; it is also the characteristic polynomial of the crystalline Frobenius asociated to V . We show that this crystalline Frobenius cannot be a scalar. In Sections 2 and 3 we prove the results concerning these Frobenius elements for 'd) and e f) , respectively. Section 2 is quite elementary, whereas in Section 3 we use a lot of the machinary for comparing " -adic etale and crystalline cohomology. In Section 4 we give some applications: the Ramanujan inequality is a strict inequality in certain cases, certain Hecke algebras are reduced, hence have non-zero discriminant. Section 5 gives some results, due to Abbes and Ullmo, concerning the discriminants of certain Hecke algebras. g partially supported by the Institut Universitaire de France

Deformation of compact Clifford-Klein forms of indefinite-Riemannian homogeneous manifolds

Abstract: 1 .1 . Let G be a Lie group, and H a closed subgroup of G . If a discrete subgroup Γ of G acts properly discontinuously and freely on G/H , then the double coset space Γ\G/H carries naturally a manifold structure such that the quotient map G/H → Γ\G/H is locally diffeomorphic. The manifold Γ\G/H is said to be a Clifford-Klein form of G/H . If it is compact, then Γ is said to be a uniform lattice for G/H . A typical example is a compact Riemann surface Mg with genus g ≥ 2, which is biholomorphic to a compact Clifford-Klein form of the Poincare disk G/H ' SL(2,R)/SO(2) by the uniformization theorem. It is important from geometric view point that a Clifford-Klein form Γ\G/H inherits any G-invariant local geometric structure on G/H such as (indefinite)Riemannian metric, complex structure, symplectic structure, causal structure and so on.

Amenable isometry groups of Hadamard spaces

Abstract: One of the ever recurring themes in the theory of nonpositively curved spaces is the relation between geometry and topology. Since the universal covering space of a complete space of nonpositive curvature is contractible, this amounts to relations between the geometry of the space (or its universal covering space) and the algebraic structure of its fundamental group. A paradigm is the well-known result of Avez [Av] that the fundamental group of a compact Riemannian manifold M of nonpositive sectional curvature has exponential growth if and only if M is not flat. Gromov observed that the proof of Avez yields a somewhat stronger statement: the fundamental group of M is amenable if and only if M is flat, see [G1, p. 93]. Zimmer generalized this to the case where M is complete and of finite volume [Zi]. Further generalizations were obtained by Anderson [An] and Burger and Schroeder [BS]. The objective of this paper is the discussion of these results under the weaker assumption that the underlying spaces are Alexandrov spaces of nonpositive curvature. For the investigation of relations between the fundamental group and the geometry of a space of nonpositive curvature, it is convenient to study the isometric action of the fundamental group on the universal covering space. In many situations, it is not necessary to assume that a group action arises in this particular way and one studies isometric actions on complete simply connected spaces of nonpositive curvature. This is the case in most of our arguments below. Let X be a Hadamard space, that is, a complete simply connected geodesic space of nonpositive curvature in the sense of Alexandrov. A flat in X is a

The convex hull of the integer points in a large ball

Abstract: where fk (P ) denotes the number of k–dimensional faces of the polytope P . The limit, as R → ∞, of the average of r−2/3f0(Pr ), on an interval [R,R + H ], is determined by Balog and Deshoullier [BD], and turns out to be 3.453 . . . , (H must be large). Our main result extends (1.1) to any d ≥ 2 and to any fk (Pr ) with k = 0, . . . , d − 1. Theorem 1. For every d ≥ 2 there are constants c1(d ) and c2(d ) such that for all k ∈ {0, . . . , d − 1}

Equations of (1, d)-polarized abelian surfaces

Abstract: In this paper, we study the equations of projectively embedded abelian surfaces with a polarization of type (1, d ). Classical results say that given an ample line bundle L on an abelian surface A, the line bundle L ⊗n is very ample for n ≥ 3, and furthermore, in case n is even and n ≥ 4, the generators of the homogeneous ideal IA of the embedding of A via L ⊗n are all quadratic; a possible choice for a set of generators of IA are the Riemann theta relations. On the other hand, much less is known about embeddings via line bundles L of type (1, d ), that is line bundles L which are not powers of another line bundle on A. It is well-known that if d ≥ 5, and A is a general abelian surface, then L is very ample, while L can never be very ample for d < 5. However, even if d ≥ 5, L may not be very ample for special abelian surfaces. We will restrict our attention in what follows only to the general abelian surface and wish to know what form the equations take for such a projectively embedded abelian surface. A few special cases are well-documented in the literature: d = 4, in which case the general surface is a singular octic in P3, cf. [BLvS], and d = 5 in which case the abelian surface is described as the zero set of a section of the Horrocks-Mumford bundle [HM], whereas its homogeneous ideal is generated by 3 (Heisenberg invariant) quintics and 15 sextics (cf. [Ma]). Also, recent work by Manolache and Schreyer [MS] and by Ranestad [Ra] provides a description of the equations and syzygies in the case d = 7.

Seshadri constants and periods of polarized abelian varieties

Abstract: The purpose of this paper is to study the Seshadri constants of abelian varieties. Consider a polarized abelian variety (A,L) of dimension g over the field of complex numbers. One can associate to (A,L) a real number e(A,L), its Seshadri constant, which in effect measures how much of the positivity of L can be concentrated at any given point of A. The number e(A,L) can be defined as the rate of growth in k of the number of jets that one can specify in the linear series |OA(kL)|. Alternatively, one considers the blow-up f : X = Blx(X) −→ X of X at a point x with exceptional divisor E ⊂ X over x, and defines e(A,L) =def sup{ e ∈ R | f ∗L− eE is nef } .

How to shell a monoid

Abstract: For a finitely generated submonoid Λ of N, we consider the minimal free resolution of a field k as a module over the monoid algebra k[Λ]. Interpreting the ranks of the free modules in the resolution as the homology of certain simplicial complexes associated to posets, we show how non-commutative Gröbner bases and the non-pure shellings of Björner and Wachs can be used to obtain information about the resolution. Complete results are obtained for monoids Λ which support posets in a certain sense.

The oka-grauert principle without induction over the base dimension

Abstract: We give a new proof of Grauert’s theorem on Oka’s principle [Gra1, Gra2, Gra3] (see also [C]) in the case of (smooth) Stein manifolds, which does not use induction over the base dimension Instead we use induction over the levels of a strictly plurisubharmonic exhausting function (Grauert’s bump method) The present paper is an edited version of our preprint [H-L2] from 1986 (which is difficult to find and of bad printing quality) We did not publish the paper in a journal at that time, because we planned to write a book containing it But the book has not been written until now On the other hand, in the meantime, some interest to this proof appeared (see, eg, the work of Gromov [Gro]) Therefore we think a publication of our proof could be useful even with a delay of 10 years This proof gives also some Oka principle for arbitrary pseudoconvex manifolds (see Theorem 13 below) For vector bundles (vector bundles without further mention will refer to topological vector bundles with fibre Cr ) this Oka principle contains the following

Complete noncompact self-similar solutions of gauss curvature flows. i. positive powers

Abstract: We classify all complete noncompact embedded convex hypersurfaces in $\\mathbf{R}^{n+1}$ which move homothetically under flow by some negative power of their Gauss curvature.

A perturbation result for prescribing mean curvature

Abstract: In this article we consider a boundary value problem associated with the conformal deformation of metrics. We are interested in the case in which a noncompact group of conformal transformation acts on the equation so that the KazdanWarner condition gives rise to obstruction as in the Nirenberg problem. The simplest situation is the following: Let Bn+1 be the unit ball in Rn+1 with Euclidean metric g0. Its boundary will be denoted by S n = ∂Bn+1. Let h be a smooth function on S n , we study the problem of finding a conformal metric gu = u4/(n−1)g0 whose scalar curvature vanishes in Bn+1 and on the boundary ∂Bn+1 the mean curvature is given by h . This problem is equivalent to solving the following boundary value problem.