On Algebraic Isomorphisms of the Rational Cohomology of Künnemann’s Compactifications of the Néron Model of an Abelian Variety without Complex Multiplication

Mathematical Notes -     

The Néron component series of an abelian variety

We introduce the Néron component series of an abelian variety A over a complete discretely valued field. This is a power series in ${\mathbb{Z}}\left[\left[T\right]\right]$ , which measures the behaviour of the number of components of the Néron model of A under tame ramification of the base field. If A is tamely ramified, then we prove that the Néron component series is rational. It has a pole at T = 1, whose order equals one plus the potential toric rank of A. This result is a crucial ingredient of our proof of the motivic monodromy conjecture for abelian varieties. We expect that it extends to the wildly ramified case; we prove this if A is an elliptic curve, and if A has potential purely multiplicative reduction.     

On the Extension of Onsager’s Conjecture for General Conservation Laws

Journal of Nonlinear Science - The aim of this work is to extend and prove the Onsager conjecture for a class of conservation laws that possess generalized entropy. One of the main findings of this...     

Variational Solutions of the Pricing PIDEs for European Options in Lévy Models

Abstract

One of the fundamental problems in financial mathematics is to develop efficient algorithms for pricing options in advanced models such as those driven by Lévy processes. Essentially there are three approaches in use. These are Monte Carlo, Fourier transform and partial integro-differential equation (PIDE)-based methods. We focus our attention here on the latter. There is a large arsenal of numerical methods for efficiently solving parabolic equations that arise in this context. Especially Galerkin and Galerkin-inspired methods have an impressive potential. In order to apply these methods, what is required is a formulation of the equation in the weak sense.

The contribution of this paper is therefore to analyse weak solutions of the Kolmogorov backward equations which are related to prices of European options in (time-inhomogeneous) Lévy models and to establish a precise link between the prices and the weak solutions of these equations. The resulting relation is a Feynman–Kac representation of the solution as a conditional expectation. Our special concern is to provide a framework that is able to cover both, the common types of European options and a wide range of advanced models in which these derivatives are priced.

An application to financial models requires in particular to admit pure jump processes such as generalized hyperbolic processes as well as unbounded domains of the equation. In order to deal at the same time with the typical pay-offs that can arise, the weak formulation of the equation is based on exponentially weighted Sobolev–Slobodeckii spaces. We provide a number of examples of models that are covered by this general framework. Examples of options for which such an analysis is required are calls, puts, digital and power options as well as basket options.     

On the convergence of the Lavrent’ev method for an integral equation of the first kind with involution

The convergence in the mean-square metric of the Lavrent’ev regularizationmethod for an integral equation with involution is established. The proof of the convergence is based on studying the behavior of the resolvent of a certain integro-differential equation related to the original equation.     

On the Structure of E-unitary Covers for an Orthodox Semigroup

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On the approximate controllability of Stackelberg–Nash strategies for linear heat equations in ℝN with potentials

In this paper, we establish hierarchic control for the linear heat equation in ?^N with potentials. Our strategy is inspired by the techniques developed by Díaz and Lions [On the approximate controllability of Stackelberg-Nash strategies. In: Díaz JI, editor. Ocean circulation and pollution control mathematical and numerical investigations. Berlin: Springer; 2005. p. 17–27]; however, many new difficulties arise due to lack of compactness of Sobolev embeddings. We manage these adversities by employing similarity variables and weighted Sobolev spaces.     

On the Existence and Explicit Estimates for the Coupling Property of Lévy Processes with Drift

In this article, we first establish new criteria for the coupling property of Lévy processes with drift. The criteria are sharp for Lévy processes and Ornstein-Uhlenbeck processes with jumps, and also strengthen the recent result of Lin and Wang (Sci China Math 55:1735–1748, Theorem 1.1, 2012). Then, using the time-change technique, we derive explicit estimates for the coupling property of subordinated Brownian motions with drift. These estimates are optimal for a large class of subordinated Brownian motions.     

On a generalisation of the Skitovich–Darmois theorem for several linear forms on Abelian groups

Aequationes mathematicae - A.M. Kagan introduced a class of distributions $$\mathcal {D}_{m, k}$$ in $$\mathbb {R}^m$$ and proved that if the joint distribution of m linear forms of n independent...     

On Boundary-Value Problems for the Laplacian in Bounded Domains with Micro Inhomogeneous Structure of the Boundaries

We consider boundary-value problems with rapidly alternating types of boundary conditions. We present the classification of homogenized （limit） problems depending on the ratio of small parameters, which characterize the diameter of parts of the boundary with different types of boundary conditions. Also we study the respective spectral problem of this type.     

On the formulation of a BEM in the Bézier–Bernstein space for the solution of Helmholtz equation

This paper proposes a novel boundary element approach formulated on the Bézier-Bernstein basis to yield a geometry-independent field approximation. The proposed method is geometrically based on both computer aid design (CAD) and isogeometric analysis (IGA), but field variables are independently approximated from the geometry. This approach allows the appropriate approximation functions for the geometry and variable field to be chosen. We use the Bézier–Bernstein form of a polynomial as an approximation basis to represent both geometry and field variables. The solution of the element interpolation problem in the Bézier–Bernstein space defines generalised Lagrange interpolation functions that are used as element shape functions. The resulting Bernstein–Vandermonde matrix related to the Bézier–Bernstein interpolation problem is inverted using the Newton-Bernstein algorithm. The applicability of the proposed method is demonstrated solving the Helmholtz equation over an unbounded region in a two-and-a-half dimensional (2.5D) domain.     

On differentiability with respect to the initial data of the solution to an SDE with a Lévy noise and discontinuous coefficients

We construct a stochastic flow generated by an stochastic differential equation with its drift being a function of bounded variation and its noise being a stable process with exponent from (1,2). It is proved that the flow is non-coalescing and Sobolev differentiable with respect to the initial data. The representation for the derivative is given.     

On the analogue of a de la Vallée Poussin's problem for arcs in the complex plane

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Estimate from Below of the First Eigenvalue for the Beltrami–Laplace Operator in the Three-Dimensional Space

In this work the following theorem is proved by elementary methods. Theorem. For all congruence-subgroups of the group PSL₂ the following inequality holds: ₁ – , where is the ring of all Gaussian integers.