Serdica Mathematical Journal

Volume 43, Numbers 3−4, 2017

C O N T E N T S

A B S T R A C T S

ON A GENERALIZATION OF MARKOWITZ PREFERENCE RELATION
Valentin Vankov Iliev viliev@math.bas.bg

2010 Mathematics Subject Classification: 06A99, 54A20, 91G10.
Key words: {mean-variance theory, efficient financial portfolio, Markowitz preference relation, topological space.

Given two families u=(u_p)_{p∈ I} and v=(v_q)_{q ∈ J} of real continuous functions on a topological space X, we define a preorder R = R(u, v) on X by the condition that any member of u is an R-increasing and any member of v is an R-decreasing function. It turns out that if the topological space X is quasi-compact and sequentially compact, then any element x ∈ X is R-dominated by an R-maximal element m ∈ X: xRm. In particular, since the (n−1)-dimensional simplex is a compact subset of Rⁿ, then considering its members as portfolios consisting of n financial assets, we obtain the classical 1952 result of Harry Markowitz that any portfolio is dominated by an efficient portfolio. Moreover, several other examples of possible application of this general setup are presented.

TRUNCATED ESTIMATION METHOD AND APPLICATIONS
Tatiana V. Dogadova aurora1900@mail.ru
Marat I. Kusainov rjrltsk@gmail.com
Vyacheslav A. Vasiliev vas@mail.tsu.ru

2010 Mathematics Subject Classification: 60G25, 62M20, 60G52, 60J60, 62F12, 93E10.
Key words: Adaptive optimal prediction, dynamic systems, autoregressive processes, stochastic differential equations, stochastic differential equations with time delay, risk function; guaranteed parameter estimation.

This paper presents an estimation method of ratio type functionals by dependent sample of fixed size. This method makes it possible to obtain estimators with guaranteed accuracy in the sense of the L_2m-norm, m ≧ 1.
As an illustration, some parametric estimation problems on a time interval of a fixed length are considered. In particular, parameters of linear continuous-time and non-linear discrete-time processes are estimated. Moreover, the parameter estimation problem of non-Gaussian Ornstein--Uhlenbeck process by discrete-time observations with guaranteed accuracy is solved.
In addition to non-asymptotic properties, the limit behavior of presented estimators is investigated. It is shown that all the truncated estimators have rates of convergence of the estimators they are based upon. These estimators are used for the construction of adaptive predictors for dynamical systems with unknown parameters.\pagebreak
The problem of asymptotic efficiency of adaptive one-step predictors for stable discrete- and continuous-time processes with unknown parameters is considered. The proposed criteria of optimality are based on the loss function, defined as a linear combination of sample size and squared prediction error's sample mean. As a rule, the optimal sample size is a special stopping time.

CARACTÉRISATION D'UNE CLASSE DE CONVERGENCES DE FONCTIONS CONVEXES ET APPLICATION À LA t-ÉPI/HYPO-CONVERGENCE DE FONCTIONS CONVEXES-CONCAVES
K. El Hajioui,
D. Mentagui dri_mentagui@yahoo.fr

2010 Mathematics Subject Classification: 49J45, 49J52, 49J40, 47N10.
Key words: Fonction convexe (convexe-concave), convergences variationnelles, t-convergence, t-épi/hypo-convergence, référentiel, approximation inf (inf&\minus;sup)-convolutive, Lagrangien augmenté, convergence simple, convergence uniforme sur les bornés.

This paper focuses on a new characterization of a class of variational convergences which play a crucial role in optimization and variational analysis. When the functions under consideration are in Γ(X) or Γ(X^*) where X is a normed linear space, this characterization is given in terms of infimal-convolution approximates associated to general kernels. When we are concerned by bivariate convex-concave saddle functions, a new convergence called t-epi/hypo-convergence is introduced and characterized first in terms of generalized augmented Lagrangians and then in terms of generalized inf-sup-convolution approximates associated to specified schemes of Legendre-Fenchel partial transforms. The proofs and results considered in this paper are original and displayed within the framework of variational analysis and the duality theory.

TAUBERIAN THEOREMS FOR THE MEAN OF LEBESGUE-STIELTJES INTEGRALS
Hristo S. Sendov hssendov@stats.uwo.ca,
Junquan Xiao jxiao48@uwo.ca

2010 Mathematics Subject Classification: Primary 40A05, 40A10, 40E05; Secondary 28A25.
Key words: Tauberian theorem, Lebegue-Stieltjes integral, statistical limit, slowly decreasing sequence.

Suppose s(x):[a, ∞) ↦ R is locally integrable with respect to a Radon measure μ on [a,∞). The mean of s(x) with respect to μ is defined to be
τ(t)=1/F(t) ∫_a^ts(x) μ (dx),
where F(x) = μ(a,x]. A scallar l is called the statistical limit of s(x) as x→ ∞ if for every ε > 0,
lim_{b → ∞} {1/(b-a) |{x ∈ (a,b) : |s(x)-l| > ε}|=0.
This is denoted by st-lim_{x → ∞} s(x)=l. The following Tauberian theorems are proved under mild assymptotic conditions on F(t) and assuming that s(x) is slowly decreasing with respect to F(t).

EUCLIDEAN SUBMANIFOLDS WITH INCOMPRESSIBLE CANONICAL VECTOR FIELD
Bang-Yen Chen chenb@msu.edu

2010 Mathematics Subject Classification: 53A07, 53C40, 53C42.
Key words: Euclidean submanifold, canonical vector field, conservative vector field, incompressible vector field.

For a submanifold M in a Euclidean space E^m, the tangential component x^T of the position vector field x of M is the most natural vector field tangent to the Euclidean submanifold, called the \textit{canonical vector field} of M. In this article, first we prove that the canonical vector field of every Euclidean submanifold is always conservative. Then we initiate the study of Euclidean submanifolds with incompressible canonical vector fields. In particular, we obtain the necessary and sufficient conditions for the canonical vector field of a Euclidean submanifold to be incompressible. Further, we provide examples of Euclidean submanifolds with incompressible canonical vector field. Moreover, we classify planar curves, surfaces of revolution and hypercylinders with incompressible canonical vector fields.

METHODS FOR CONSTRUCTING FACTORIZATIONS OF ABELIAN GROUPS WITH APPLICATIONS
S. Szabó sszabo7@hotmail.com

2010 Mathematics Subject Classification: 53A07, 53C42.
Key words: Dupin hypersurfaces, Lie curvature, Laplace invariants, lines of curvature.

In this paper we study Dupin hypersurfaces in R⁵ parametrized by lines of curvature, with four distinct principal curvatures. We give a local characterization of this class of hypersurfaces in terms of the principal curvatures and four vector valued functions of one variable. We prove that these vectorial functions describe plane curves or points in R⁵. We show that the Lie curvature of these Dupin hypersurfaces is constant with some conditions on the Laplace invariants and the Möbius curvature, but some Möbius curvatures are constant along certain lines of curvature. We give explicit examples of such Dupin hypersurfaces.

R. Dangovski rumenrd@mit.edu,
C. Lalov chavdar.lalov@gmail.com

2010 Mathematics Subject Classification: 05A15, 05C38, 05C81, 60G50, 82B41.
Key words: self-avoiding walks, connective constant, honeycomb lattice, asymptotic behaviour.

We study self-avoiding walks (SAWs) on restricted square lattices, more precisely on the lattice strips Z × {−1, 0, 1} and Z × {−1, 0, 1, 2}. We obtain the value of the connective constant for the Z × {−1, 0, 1} lattice in a new shorter way and deduce close bounds for the connective constant for the Z × {−1, 0, 1, 2} lattice. Moreover, for both lattice strips we find close lower and upper bounds for the number of SAWs of length n by using the connective constant. Additionally, we present a transformation of SAWs on the square lattice to a special kind of walks on the honeycomb lattice. By using H. Duminil-Copin and S. Smirnov's results for SAWs on the honeycomb lattice we present non-rigorous ways by which close bounds for the number of SAWs and for the connective constant of the non-restricted square lattice could eventually be obtained without the need of long computer computations.