Showing papers in "Journal of Applied Mathematics and Stochastic Analysis in 2002"
TL;DR: In this article, the existence, uniqueness and continuous dependence on the data of a solution of a mixed problem with a weighted integral condition for a parabolic equation with the Bessel operator were proved.
Abstract: In this paper, we prove the existence, uniqueness and continuous dependence on the data of a solution of a mixed problem with a weighted integral condition for a parabolic equation with the Bessel operator. The proof uses a functional analysis method based on an a priori estimate and on the density of the range of the operator generated by the considered problem. Singular Parabolic Equations, Weighted Integral Condition, A
30 citations
27 citations
TL;DR: In this paper, the authors study the finite delay evolution equation with non-densely defined linear operators and prove the existence of periodic integral solutions when integral solutions are bounded and ultimately bounded.
Abstract: We study the finite delay evolution equation { x ' ( t ) = A x ( t ) + F ( t , x t ) , t ≥ 0 , x 0 = ϕ ∈ C ( [ − r , 0 ] , E ) ,
where the linear operator A is non-densely defined and satisfies the Hille-Yosida
condition. First, we obtain some properties of integral solutions for this case
and prove the compactness of an operator determined by integral solutions. This
allows us to apply Horn's fixed point theorem to prove the existence of periodic
integral solutions when integral solutions are bounded and ultimately bounded.
This extends the study of periodic solutions for densely defined operators to the
non-densely defined operators. An example is given.
23 citations
TL;DR: In this paper, the authors prove the existence, uniqueness and continuous dependence of a class of nonclassical hyperbolic equations with nonlocal boundary conditions and initial conditions by using a functional analysis method.
Abstract: This paper proves the existence, uniqueness and continuous dependence of a
solution of a class of nonclassical hyperbolic equations with nonlocal boundary
and initial conditions. Results are obtained by using a functional analysis method
based on an a priori estimate and on the density of the range of the linear operator
corresponding to the abstract formulation of the considered problem.
21 citations
TL;DR: In this article, various formulas for the Laplace transform of Gauss-Markov sequences have been derived, and relationships between the solutions of matched forward and backward Riccati equations are proved probabilistically; they are proved again directly.
Abstract: Various methods to derive new formulas for the Laplace transforms of some quadratic forms of Gaussian sequences are discussed. In the general setting, an approach based on the resolution of an appropriate auxiliary filtering problem is developed; it leads to a formula in terms of the solutions of Voterra type recursions describing characteristics of the corresponding optimal filter. In the case of Gauss-Markov sequences, where the previous equations reduce to ordinary forward recursive equations, an alternative approach provides another formula; it involves the solution of a backward recursive equation. Comparing the different formulas for the Laplace transform- s, various relationships between the corresponding entries are identified. In particular relationships between the solutions of matched forward and backward Riccati equations are thus proved probabilistically; they are proved again directly. In various specific cases, a further analysis of the concerned equations leads to completely explicit formulas for the Laplace transform.
20 citations
TL;DR: In this article, a stochastic generalization of some fixed point theorems on a class of nonconvex sets in a locally bounded topological vector space is established.
Abstract: Stochastic generalizations of some fixed point theorems on a class of nonconvex
sets in a locally bounded topological vector space are established. As applications, Brosowski-Meinardus type theorems about random invariant approximation are obtained. This work extends or provides stochastic versions of several well known results.
16 citations
TL;DR: In this paper, sufficient coefficient conditions for complex functions to be close-to-convex harmonic or convex harmonic are given, and a convolution property for harmonic functions is discussed.
Abstract: Sufficient coefficient conditions for complex functions to be close-to-convex harmonic
or convex harmonic are given. Construction of close-to-convex harmonic
functions is also studied by looking at transforms of convex analytic functions.
Finally, a convolution property for harmonic functions is discussed.
14 citations
TL;DR: The convective diffusion equation with drift b(x) and indefinite weight r(x), ∂ϕ∂t=∂∂x[a∂ ϕ ∂x−b(x))ϕ]
Abstract: The convective diffusion equation with drift b(x) and indefinite weight r(x), ∂ϕ∂t=∂∂x[a∂ϕ∂x−b(x)ϕ]
12 citations
TL;DR: In this paper, the authors derived large deviation principles for a spatially coarse-grained process with respect to either the canonical and/or the microcanonical formulation of the model.
Abstract: Statistical equilibrium lattice models of coherent structures in geostrophic turbulence, formulated by discretizing the governing Hamiltonian continuum dynamics, are analyzed. The first set of results concern large deviation principles (LDP's) for a spatially coarse-grained process with respect to either the canonical and/or the microcanonical formulation of the model. These principles are derived from a basic LDP for the coarse-grained process with respect to product measure, which in turn depends on Cramer's Theorem. The rate functions for the LDP's ´ give rise to variational principles that determine the equilibrium solutions of the Hamiltonian equations. The second set of results addresses the equivalence or nonequivalence of the microcanonical and canonical ensembles. In particular, necessary and sufficient conditions for a correspondence between microcanonical equilibria and canonical equilibria are established in terms of the concavity of the microcanonical entropy. A complete characterization of equivalence of ensembles is deduced by elementary methods of convex analysis. The mathematical results proved in this paper complement the physical reasoning and numerical computations given in a companion paper, where it is argued that the statistical equilibrium model defined by a prior distribution on potential vorticity fluctuations and microcanonical conditions on total energy and circulation is natural from the perspective of geophysical applications. Large Deviation Principle, Cramer's Theorem, Coarse Graining,
12 citations
TL;DR: In this paper, the authors considered planar random motions with four directions and four different speeds, switching at Poisson paced times, and obtained the explicit distribution of the position (X(t),Y(t)), t>0 in all its components (the discrete one, lying on the edge and the absolutely continuous one, concentrated inside Qt).
Abstract: In this paper we consider planar random motions with four directions and four different speeds, switching at Poisson paced times. We are able to obtain, in some cases, the explicit distribution of the position (X(t),Y(t)), t>0 in all its components (the discrete one, lying on the edge ∂Qt of the probability support Qt, as well as the absolutely continuous one, concentrated inside Qt).
11 citations
TL;DR: In this paper, the authors considered the MAP/G/1 queue under N-policy with a single vacation and set-up and derived the vector generating functions of the queue length at an arbitrary time and at departures in decomposed forms.
Abstract: This paper considers the MAP/G/1 queue under N-policy with a single vacation and set-up. We derive the vector generating functions of the queue length at an arbitrary time and at departures in decomposed forms. We also derive the Laplace-Stieltjes transform of the waiting time. Computation algorithms for mean performance measures are provided.
TL;DR: In this paper, a fixed point theorem due to Schaefer is used to investigate the existence of solutions for second order impulsive functional differential equations in Banach spaces, and the authors show that this theorem is correct.
Abstract: In this paper, a fixed point theorem due to Schaefer is used to investigate the
existence of solutions for second order impulsive functional differential equations
in Banach spaces.
TL;DR: In this article, the invariance of domain for contractive field and Schauder invertibility theorem is proved for open embeddings, and the stability of Probabilistic Open Embedding is established.
Abstract: Probabilistic version of the invariance of domain for contractive field and
Schauder invertibility theorem are proved. As an application, the stability of
probabilistic open embedding is established.
TL;DR: In this paper, the multi-dimensional oscillation problem is reduced to a one-dimensional problem for higher order functional differential inequalities, and sufficient conditions are derived for every solution of certain boundary value problems to be oscillatory in a cylindrical domain.
Abstract: Higher order partial differential equations with functional arguments including hyperbolic equations and beam equations are studied. Sufficient conditions are derived for every solution of certain boundary value problems to be oscillatory in a cylindrical domain. Our approach is to reduce the multi-dimensional oscillation problem to a one-dimensional problem for higher order functional differential inequalities.
TL;DR: In this article, the authors prove the existence of solutions of nonlinear second order integrodifferential Equations in Banach spaces by using the theory of strongly continuous cosine families of operators and the Schaefer fixed point theorem.
Abstract: In this paper we prove the existence of solutions of nonlinear second order integrodifferential
equations in Banach spaces. The results are obtained by using
the theory of strongly continuous cosine families of operators and the Schaefer
fixed point theorem.
TL;DR: In this paper, the authors proposed a robust stability theory for systems of differential-difference equations with deviating argument of neutral type, which takes into account not only the previous moments of time, but also the speed of their change.
Abstract: Systems of differential equations with deviating argument of neutral type [1, 3, 8]
are used. The mathematical model takes into account not only the previous
moments of time, but also the speed of their change. These equations more
adequately describe the dynamics of processes, but their investigation faces
significant difficulties. The qualitative behavior of solutions of neutral type
systems includes the features both, differential and difference equations [2, 9].
Stability investigations require the smallest size of the delay's speed value [7].
Recently, so-called interval, or robust stability theory has received intensive
development. It is based on two theorems of V.L. Kharitonov [4, 5] for scalar:
equations. However, difficulties have appeared to obtain similar results for
systems in vector-matrix form. It is even more complicated to derive conditions
of interval stability for systems of differential-difference equations, though there
are results for scalar equation in [6, 10].
TL;DR: In this article, a martingale representation formula for functionals of diffusion processes with Lipschitz coefficients was established, as stochastic integrals with respect to the Brownian motion.
Abstract: We establish a martingale representation formula for functionals of diffusion processes with Lipschitz coefficients, as stochastic integrals with respect to the Brownian motion.
TL;DR: In this paper, it was shown that both the Statonovich and Ito s calculi lead to the same reactive Fokker-Planck equation: ∂p∂t−∂∂x[D∂p ∂x−bp]=λmp, describing stochastic dynamics of a particle moving under the influence of an indefinite potential m(x,t), a drift b(x and t), and a constant diffusion D.
Abstract: In this paper, we show that despite their distinction, both the Statonovich and Ito s calculi lead to the same reactive Fokker-Planck equation: ∂p∂t−∂∂x[D∂p∂x−bp]=λmp, (1) describing stochastic dynamics of a particle moving under the influence of an indefinite potential m(x,t), a drift b(x,t), and a constant diffusion D. We treat the periodic-parabolic eigenvalue problem (1) for finite domains having absorbing barriers. We show that under conditions required by the maximum principle, the positive principal eigenvalue λ* (and the negative principal λ* eigenvalue) is connected to the probability eigendensity function p(x,t) by a Raleigh-Ritz like formulation. In the process, we establish the manner of effect of the drift and any inducing potential on the size of the principal eigenvalue. We show that the degree of convexity of the potential plays a major role in this regard.
TL;DR: In this paper, the estimation problem of the expected local fraction of free area function S for a partially observed dynamic germ-grain model is presented, and confidence bounds are provided by martingale and product integral methods.
Abstract: The estimation problem of the expected local fraction of free area function S for a
partially observed dynamic germ-grain model is presented. Properties of the
estimators are proved by martingale and product integral methods. Confidence
bounds are provided. Furthermore, an estimator of the hazard rate
α ( t ) = − d S ( t ) / ( S ( t ) d t )
is obtained by the kernel function method and
asymptotic properties of the estimator are proved and used to find confidence
intervals. By a simulated illustrative example, the qualitative behavior of the
estimators is shown.
TL;DR: In this article, a class of hybrid LQG (linear quadratic Gaussian) Regulator problems modulated by continuous-time Markov chains is considered, where the Markov chain is formulated as singularly perturbed with a small parameter.
Abstract: This work is concerned with a class of hybrid LQG (linear quadratic Gaussian)
regulator problems modulated by continuous-time Markov chains. In contrast to
the traditional LQG models, the systems have both continuous dynamics and
discrete events. In lieu of a model with constant coefficients, these coefficients
vary with time and exhibit piecewise constant behavior. At any time t, the system
follows a stochastic differential equation in which the coefficients take one of the
m
possible configurations where m
is usually large. The system may jump to any
of the possible configurations at random times. Further, the control weight in the
cost functional is allowed to be indefinite. To reduce the complexity, the Markov
chain is formulated as singularly perturbed with a small parameter. Our effort is
devoted to solving the limit problem when the small parameter tends to zero via
the framework of weak convergence. Although the limit system is still modulated
by a Markov chain, it has a much smaller state space and thus, much reduced
complexity.
TL;DR: In this article, a generalized Jacobi transform was studied and images of certain functions under this transform were obtained, and a Jacobi random variable was defined and its moments, distribution function, and characteristic function were derived.
Abstract: In this paper, we study a generalized Jacobi transform and obtain images of
certain functions under this transform. Furthermore, we define a Jacobi random
variable and derive its moments, distribution function, and characteristic function.
TL;DR: In this article, the ordinary H∞-control theory is extended to locally convex spaces through the form of a parameter, and the algorithms of computing the infimal model-matching error and infimal controller are presented.
Abstract: In this paper, the ordinary H∞-control theory is extended to locally convex spaces through the form of a parameter. The algorithms of computing the infimal model-matching error and the infimal controller are presented in a locally convex space. Two examples with the form of a parameter are enumerated for computing the infimal model-matching error and the infimal controller.
TL;DR: In this article, the authors studied the issue of strong consistency for two different slope estimators: the least square estimator and the weighted least squares estimator, assuming the unobserved disturbance e i are negatively associated.
Abstract: In this paper we are concerned with the heteroscedastic regression model y i = x i β + g ( t i ) + σ i e i , 1 ≤ i ≤ n under correlated errors e i , where it is assumed that σ i 2 = f ( u i ) , the design points ( x i , t i , u i ) are known and nonrandom, and g and f are unknown functions. The interest lies in the slope parameter β . Assuming the unobserved disturbance e i are negatively associated, we study the issue of strong consistency for two different slope estimators: the least squares estimator and the weighted least squares estimator.
TL;DR: In this article, the approximate solvability of a class of nonlinear implicit variational inequalities with partially relaxed monotone mappings has been investigated in a Hilbert space setting with some applications.
Abstract: Approximation-solvability of a class of nonlinear implicit variational inequalities
involving a class of partially relaxed monotone mappings - a computation-oriented
class in a Hilbert space setting- is presented with some applications.
TL;DR: In this paper, an upper bound for the uniform distance between distributions of sums of a random number of independent and identically distributed random variables is established, and the right hand side of these inequalities are expressed in terms of Zolotarev's and the uniform distances between the distributions of summands.
Abstract: The paper deals with sums of a random number of independent and identically distributed random variables. More specifically, we compare two such sums, which differ from each other in the distributions of their summands. New upper bounds (inequalities) for the uniform distance between distributions of sums are established. The right-hand sides of these inequalities are expressed in terms of Zolotarev's and the uniform distances between the distributions of summands. Such a feature makes it possible to consider these inequalities as continuity estimates and to apply them to the study of the stability (continuity) of various applied stochastic models involving geometric sums and their generalizations.
TL;DR: In this paper, the authors provided an asymptotic value for the mathematical expected number of points of inflections of a random polynomial in the form of a 0 ( ω ) + a 1( ω) (n 1 ) 1/2 x + a 2 ( ψ ) (n 2 ) 1 / 2 x 2 + … a n (π) ( n n ) 1 2 /2 x n when n is large.
Abstract: This paper provides an asymptotic value for the mathematical expected number of
points of inflections of a random polynomial of the form
a 0 ( ω ) + a 1 ( ω ) ( n 1 ) 1 / 2 x + a 2 ( ω ) ( n 2 ) 1 / 2 x 2 + … a n ( ω ) ( n n ) 1 / 2 x n when n is large. The coefficients { a j ( w ) } j = 0 n , w ∈ Ω are assumed to be a sequence of independent
normally distributed random variables with means zero and variance one, each
defined on a fixed probability space ( A , Ω , Pr ) . A special case of dependent coefficients is also studied.
TL;DR: In this paper, a fixed point theorem for condensing maps combined with upper and lower solutions is used to investigate the existence of solutions for first order functional differential inclusions, and the results are shown to be equivalent to the results in this paper.
Abstract: In this paper, a fixed point theorem for condensing maps combined with upper and lower solutions are used to investigate the existence of solutions for first order functional differential inclusions.
TL;DR: In this paper, the notion of Lipschitz stability was introduced for nonlinear n th order matrix Lyapunov differential systems and sufficient conditions for Lipschnitz stability were given.
Abstract: This paper introduces the notion of Lipschitz stability for nonlinear n th order
matrix Lyapunov differential systems and gives sufficient conditions for Lipschitz
stability. We develop variation of parameters formula for the solution of the
nonhomogeneous nonlinear n th order matrix Lyapunov differential system. We
study observability and controllability of a special system of n th order nonlinear Lyapunov systems.