Showing papers in "Random Operators and Stochastic Equations in 2006"

Extrapolation of multidimensional stationary processes

Abstract: Estimation problems are considered for a functional which depends on the unknown values of a multidimensional stationary stochastic process based on observations of the process for t < 0. Formulas are proposed for calculation the mean square error and the spectral characteristics of the optimal estimate of the functional under the condition that the spectral density of the process is known. The least favorable spectral densities and the minimax spectral characteristics of the optimal estimate of the functional are found for concrete classes of spectral densities.

On uniform convergence of wavelet expansions of φ-sub-Gaussian random processes

Abstract: In the paper we present conditions for uniform convergence with probability one of wavelet expansions of φ-sub-Gaussian (in particular, Gaussian) random processes defined on the space R. It is shown that upon certain conditions for the bases of wavelets the wavelet expansions of stationary almost sure continuous Gaussian processes and wavelet expansions of fractional Brownian motion converge uniformly with probability one on any finite interval.

Necessary and sufficient conditions of optimality for optimal control problem with initial and terminal costs

Abstract: We consider a stochastic control problem where the control domain need not be convex, the system is governed by a non linear forward-backward stochastic differential equation with nonconstant terminal condition.The criteria to be minimized is in the general form, with initial and terminal costs. We derive necessary as well as sufficient conditions of optimality by introducing three adjoint equations. This problem may have applications in the financial market and it can be adapted to the problem of the minimization of an initial investment and the maximization of a final wealth.

A note on the Itô formula of stochastic integrals in Banach spaces

Abstract: Let E and Z be separable Banach spaces, E be of M type p, 1 ≤ p ≤ 2, and X = {X(t), 0 ≤ t ≤ T} be an E–valued stochastic process given by where is progressive measurable process and is a càglàd process defined later, is a compensated Poisson random measure with characteristic Lévy–measure such that We verify the Itô formula for X under certain conditions in M type p Banach spaces.

On mixing and convergence rates for a family of Markov processes approximating SDEs

Abstract: We study a class of Markov processes of the type Xn+1,h = Xn,h+ F (Xn,h)h + √ h ξn+1, where F : Rd → Rd is a bounded continuous function, (ξn) are i.i.d. random variables with zero mean, and t = nh understood as “macro-time”. Such processes are approximations to the SDE, dXt = F (Xt) dt + dWt. Upper estimates for β-mixing and convergence rates to invariant measure are established under certain assumptions on smoothness of F , the density of ξn and some recurrence conditions. The estimates are analogous to those for the limiting SDE.

The Strong Elliptical Galactic Law. Sand Clock density. Twenty years later. Part II

Abstract: We prove the strong Elliptical Galactic law for random matrices Ξ n of the general form, i.e. their diagonal entries have nonzero expectations and the pairs of the entries ( , ) have nonzero covariances. In this case the Elliptical Galactic law means that the support of the accompanying spectral density of eigenvalues of matrix Ξ n looks like the picture of several galaxies made by telescope: The picture 1 shows the collision of elliptic supports of the limit spectral density of n.s.f. of random matrix A n + Λ n Ξ n , where A n is a diagonal complex matrix with diagonal entries (0.7, 0), (−1, 0), (0, 0.7i) for corresponding three equal parts of the main diagonal, and random matrix Ξ n has equal covariances ρ(√ρ = 0.2 + i0.8) of independent pairs of entries ( , ) with zero mean and is multiplied by diagonal matrix Λ n with diagonal entries (1, 0), (0.5, 0.5i), (−1, 0) for corresponding three equal parts of the main diagonal. We have chosen in picture 1 three different diagonal entries of the matrix A n at a short distance. In picture 2, we consider the diagonal matrix A n with diagonal entries (2, 0), (−2, 0), (0, 2i) at a large distant for corresponding three equal parts of the main diagonal. In the letter case we have several domains-supports like ellipses. For the exposition of the Elliptic Law we have chosen the random matrix Ξ n of dimension 30 and 300 its Monte-Carlo simulation. If the distances between the centers of these galaxies are large enough we have several almost elliptical galaxies. These pictures show the elliptic support of the limit spectral density of n.s.f. of random matrix A n + Ξ n , where A n is a diagonal matrix with 5 different diagonal entries (1, 0); (−1, 0); (−0.5,−i); (0, 0.5i); (0, i) and random matrix Ξ n has equal covariances ρ(√ρ = 0.5 + i0.5)of the entries . We have chosen five different diagonal entries of the matrix A n at a short distance in picture 1 and at a large distant (2, 0); (−2, 0); (−1,−2i); (0, i); (0, 2i) in picture 2. In the letter case we have several domains-supports like ellipses. For the exposition of the Elliptic Law we have chosen the random matrix Ξ n of dimension 50 and 300 its Monte-Carlo simulation. If the distances between the centers of these galaxies are large enough we have several almost elliptical galaxies. Maybe the reader remembers the Monte Carlo simulations of eigenvalues of matrices Ξ n + A n , where Ξ n belongs to the domain of attraction of Circular law and A n is the diagonal matrix whose diagonal entries forms letter R on a complex plain[25]–[27]. For the case when the matrix Ξ n belongs to the domain of attraction of Elliptic law the simulation of eigenvalues of the matrix Ξ n + A n looks like the following picture: These statements are based on the VICTORIA-transform of random matrix which is the abbreviation of the following words: Very Important Computational Transformation Of Random Independent Arrays. We follow the main strategy of the theory of limit theorems of the probability theory, i.e. we try to solve the problem of description of all limits of normalized spectral functions where λ k (A n Ξ n B n + C n ) are eigenvalues of non Hermitian matrix A n Ξ n B n + C n , A n , B n , and C n are nonrandom matrices, under general (as only possible) conditions on the entries of random matrices Ξ n , χ is the indicator function. We emphasize that the spectral theory of Hermitian random matrices is rather profound theory [3,13,23,24]. There are essentially three methods of the proof of Elliptic Laws that have been proposed: the REFORM method and Berry-Esseen inequality[11], the method of perpendiculars[15,16], the method of the central limit theorem and limit theorems for eigenvalues of random matrices[23]. The main advantage of REFORM approach is that it enables the results of the previous version of Elliptic law to be extended to the case under consideration. The REFORM-method(or G-martingale approach) enables us to suggest a new method for construction of stochastic canonical equations. In this paper we prove the following Elliptical Galactic Law which generalizes the Strong Circular Law and Weak Circular Law(see the sketch of the proof of this law in the paper V-transform, Dopovidi Akademii nauk Ukrainskoi RSR, Seria A Fizyko-Matematychni ta technichni nauky, 1982, N3, pp.5-6.): For every n, let the pairs of random entries of the complex matrix be independent and given on a common probability space, i, j = 1, ..., n, and where are square complex nonrandom matrices, det A n ≠ 0, det B n ≠ 0, and the real and imaginary parts of random entries have the densities satisfying the corrected Elliptic condition: for some β > 1 or and there exist the densities of the random entries , or the densities of the random entries , satisfying the condition: for some β 1 > 1 the Lyapunov condition is fulfilled: for some δ > 0, Then, with probability one, for almost all x and y where λ k are eigenvalues of the matrix A n Ξ n B n + C n , the Global probability density p n,α (t, s) =(∂2/∂t ∂s)F n,α (t, s) is equal to where α > 0, where is a block matrix, are entries of the matrix is the block diagonal matrices, whose diagonal block satisfy the system of canonical equations K 97 , j = 1, ..., n, and is a support of the Global probability density, where There exists a unique solution of canonical equation K 97 in the class of positive definite block matrices of the order 2 × 2, analytic in y > 0, t, s.

Subordinators in a Class of Banach Spaces

Abstract: Lévy processes with values in cones of linear spaces are also called subordinators. They are studied in this paper for a class of Banach spaces called of Birkhoff-Kakutani (BK) type. We show that they share several properties with one dimensional subordinators and inherit many others from their associated random norm subordinators. This includes subordinators in the cone (H) of positive trace class operators in a separable Hilbert space H and in dual cones of Banach duals of C ∗-algebras. A general method for constructing Banach-valued subordinators in cones generated by bases is given and concrete examples of subordinators in BK spaces are presented.

Stochastic equations with multidimensional drift driven by Levy processes

Abstract: The stochastic equation dXt = dLt + a(t,Xt)dt, t ≥ 0, is considered where L is a d-dimensional Levy process with the characteristic exponent ψ(ξ), ξ ∈ IR, d ≥ 1. We prove the existence of (weak) solutions for a bounded, measurable coefficient a and any initial value X0 = x0 ∈ IR when (Reψ(ξ))−1 = o(|ξ|−1) as |ξ| → ∞. 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. AMS Mathematics subject classification. Primary 60H10, 60J60, 60J65, 60G44

Strongly dependent Gaussian scenarios for the Burgers turbulence problem with quadratic external potential

Abstract: We study the limiting distributions of a rescaling solution of the heat and Burgers equations, with quadratic external potential, in the case where the initial velocity potential is a strongly dependent Gaussian random field.

Studying anticipation on financial markets by BSDE

Abstract: The aim of the present survey is to give an outline of the modern mathematical tools which can be used on a financial market by a "small" investor who possesses some information on the price process (this is a higher information level which we call strong). We extensively use the backward stochastic differential equation theory to give sufficient condition to compare the strategies of an insider trader and the non insider one.

Limiting distribution of random motion in a n-dimensional parallelepiped

Abstract: In this paper we study a continuous time random walk in a n-dimensional parallelepiped with pairs of boundaries [ai, bi]. In a pair of boundaries the particle can move in any of two directions with different velocities v (1) i and v (2) i . 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.

On a generalized BSDE involving local time and application to a PDE with nonlinear boundary condition

Abstract: We consider the following generalized BSDE: where (Bt , 0 ≤ t ≤ T) is a d-dimensional Brownian motion, ξ is the terminal value, {kt , 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.

Comparative stationarity of stochastic exponential and monomial densities

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.

On Poisson and Gaussian measures in Euclidian spaces

Abstract: We consider random measures as measures with values in the Hilbert space of random variables on some probability space see [1,2]. A Poisson measure is the family of random variables satisfying the conditions (P1) for any finite sequence for which the random variables π(Ak ) are independent , (P2) the random variable π(A) has a Poisson distribution with the parameter m(A) where m is a measure on B(Rd ) . The main result of the article concerning Poisson measures is the proof of the existence of a sequence {x k (ω), k ≥ 1} of R d-valued random variables for which A random measure is Gaussian if for any finite sequence {Ak ∈ B(Rd ), k ≤ n} the random variables {μ(Ak ), k ≤ n} have joint Gaussian distribution. As a rule measures were considered under the additional assumption that the values are independent for disjoint subsets. We consider the existence of a measure in the general case. Introduce the correlation function It is proved that any positively defined function is the correlation function for some Gaussian measure.

Transformation of Gaussian measure by infinite-dimensional stochastic flow

Abstract: We study the propagation of absolute continuity of Gaussian measure in inflnite- dimensional space by a ∞ow generated by stochastic equation.

On existence of a solution for differential equation with interaction

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.

Complex generalized binomial coefficients

Abstract: In this article we define generalized binomial coefficients associated with zonal polynomials of Hermitian matrix arguments, study their properties and other results.