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Nassar H. S. Haidar, Adnan M. Hamzeh, Soumaya M. Hamzeh
December 1, 2006

### Abstract

Stochastic exponential and monomial densities can serve as important sets of base functions both in linear and nonlinear functional analysis. The present paper is an extension of a work [1] of the first author to study the comparative stationarity of stochastic exponential densities that are compactly supported over [a,b] in R.

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V. P. Kurenok
December 1, 2006

### Abstract

The stochastic equation dX t = dL t + a ( t,X t ) dt , t > 0, is considered where L is a d-dimensional Levy process with the characteristic exponent (ξ), ξ ∈ Bbb R, d > 1. We prove the existence of (weak) solutions for a bounded, measurable coefficient a and any initial value X 0 = x 0 ∈ when The proof idea is based on Krylov's estimates for Levy processes with time-dependent drift and some variants of those estimates are derived in this note.

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V. B. Brayman
December 1, 2006

### Abstract

Consider the following differential equation with interaction The conditions that guarantee the existence of a solution are obtained when mapping a belongs to some Sobolev space as a function of first coordinate.

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Taras Androshchuk
December 1, 2006

### Abstract

We give sufficient conditions under which dynamical system with small fractional Brownian noise generates a set of regular statistical experiments in a sense of Ibragimov and Has'minskii's definition. As a corollary, the maximum likelihood estimator of unknown parameter based on the observation of trajectory is consistent, uniformly asymptotically normal and its moments converge to the ones of the standard normal distribution.

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B. Boufoussi, N. Mrhardy
December 1, 2006

### Abstract

We consider the following generalized BSDE: where ( B t , 0 ≤ t ≤ T ) is a d-dimensional Brownian motion, ξ is the terminal value, { k t , 0 ≤ t ≤ T } is a continuous real valued increasing process such that k 0 = 0, ν is a signed measure on and is the symmetric local time of the semimartingale Y . Under some continuous and linear growth conditions on the coefficients ƒ and h , we will prove existence result for equation of the type (1). As a consequence we will give a probabilistic representation to the solution of a nonlinear partial differential equations with Neumann boundary conditions.

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A. A. Pogorui, Ramón M. Rodríguez-Dagnino
December 1, 2006

### Abstract

In this paper we study a continuous time random walk in a n-dimensional parallelepiped with pairs of boundaries [ a i , b i ]. In a pair of boundaries the particle can move in any of two directions with different velocities and We consider a special type of boundary which can trap the particle for a random time, and we found the limiting distribution of this random motion for the position of the particle. Our formulation allows us to find the limiting distribution for a broad class of alternating semi-Markov processes.

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Mikhail P. Moklyachuk, Aleksey G. Zrazhevsky
December 1, 2006

### Abstract

The problem of estimation of the Hurst parameter for self-similar time series is discussed in the paper. Five methods of estimation of the Hurst parameter for prices of MSFT ticker, for returns of MSFT ticker and for simulated FARIMA time series with H = 0.766 are presented. Methods that are inefficient for estimation the Hurst parameter in limit cases ( H close to 0.5 and H close to 1) are detected based on the presented methods. The long-range dependence of the mentioned three time series are statistically proved.