Asymptotic behavior of solutions of stochastic recurrence equations in Rd

We study convergence to zero and boundedness with probability one of solutions of stochastic recurrence equations under matrix norms.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 6, pp. 829–833, 1991.     

Asymptotic behavior of solutions of stochastic functional-differential equations with Poisson switchings

We obtain conditions of global asymptotic stability of solutions of stochastic functional-differential equations with Poisson switchings. Translated from Ukrainskii Matematicheskii Zhurnal, Vol.50, No. 6, pp. 845–861, June, 1998.     

Asymptotic behavior of solutions of stochastic evolution equations for second grade fluids

In this Note we show that under suitable conditions on the data we can construct a sequence of solutions of the stochastic second grade fluid that converges to the probabilistic strong solution of the stochastic Navier–Stokes equations when the stress modulus α tends to zero.     

Asymptotic behavior of solutions of some classes of stochastic equations and their applications to statistical problems

Conditions of stability and asymptotic normality are derived for solutions of equations of form . Heref(, ·) is a family of functions and ( _k , _k ) is a sequence of conditionally independent variables inR ^r given on x _k , k1, where x _k is a sequence with values in an arbitrary space that satisfies a certain ergodicity condition. The continuous-time case is also considered. The asymptotic properties of maximum likelihood estimators and moment method estimators are investigated for observation of weakly dependent variables.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 57, pp. 103–112, 1985.     

Asymptotic behavior of continuous stochastic games

Summary The aim of this paper is to analyze the asymptotic behavior of the value functions of a continuous stochastic game as the number of stages grows to infinity or the discount factor approaches 1. After the setup of the problem we prove that, in both cases, the extrema of the value functions converge to the same limits. The convergence of the value functions is then obtained from the unicity of the solution of a functional problem and it is thus possible to design hypotheses that assure the convergence to a constant. This allows to assign a value to an undiscounted infinite-stage stochastic game in several senses and to show that optimal strategies are available for both players. Furthermore the boundedness of the remainders of the value function after removing the principal terms is analyzed, with appropriate hypotheses, and related to the existence of solutions of a Howard's type functional equation. This allows to show that for an infinite-stage undiscounted stochastic game optimal stationary strategies exist at least if this functional equation has some solution.     

Asymptotic behavior of solutions to polynomial renewal equations

Many special functions arise as “renormalized” limits of sequences of polynomials that satisfy a polynomial renewal equation. We determine the asymptotic behavior of these sequences of polynomials for both ordinary and “coefficientwise” convergence, and illustrate it with specific examples.     

Asymptotic behavior of truncated stochastic approximation procedures

We study asymptotic behavior of stochastic approximation procedures with three main characteristics: truncations with random moving bounds, a matrix-valued random step-size sequence, and a dynamically changing random regression function. In particular, we show that under quitemild conditions, stochastic approximation procedures are asymptotically linear in the statistical sense, that is, they can be represented as weighted sums of random variables. Therefore, a suitable formof the central limit theoremcan be applied to derive asymptotic distribution of the corresponding processes. The theory is illustrated by various examples and special cases.     

Asymptotic behavior of solutions to nonlinear Volterra integral equations

A class of operator Riccati integral equations is associated with a factorization problem in a certain Banach algebra. Recent results concerning factorization in this algebra are used to obtain existence, uniqueness, and continuous dependence results for the Riccati equations.     

Asymptotic behavior of solutions to abstract functional differential equations

In this paper, a new approach is provided to study the asymptotic behavior of functions. A Tauberian theorem is improved and applied to describe the asymptotic behavior of abstract functional differential equations of the form

     

Asymptotic behavior of impulsive stochastic functional differential equations

In this paper, a nonlinear and nonautonomous impulsive stochastic functional differential equation is considered. By establishing a nonautonomous -operator impulsive delay inequality and using the properties of ρ-cone and stochastic analysis technique, we obtain the p-attracting set and p-invariant set of the impulsive stochastic functional differential equation. An example is also discussed to illustrate the efficiency of the obtained results.     

Asymptotic properties of solutions of multidimensional stochastic differential equations

Summary LetX _t R ^d be the solution of the stochastic equationdX _t =b(X _t )dt+(X _t )dW _t , whereW _t denotes a standard Wiener process. The aim of the paper is to clarify under which conditions the drift term or the diffusion term is of negligible significance for the long term behaviour ofX _t .     

Asymptotic behavior of solutions of variational equations

In questo articolo si considerano equazioni differenziali ordinarie del secondo ordine della forma $$\{ A(u')u'\} ' + \delta (r)A(u')u' + f(r,u) = 0,$$ dove cioè la nonlinearità è presente sia nella variabile soluzioneu che nella sua derivata. Si forniscono proprietà di monotonia, oscillazione e un accurato studio del comportamento asintotico all’ infinito delle soluzioni, quandoA, δ,f hanno crescite asintotiche di tipo algebrico.     

Asymptotic equivalence of solutions of linear Itô stochastic systems

We investigate the problem of the asymptotic equivalence of stochastic systems of linear ordinary equations and stochastic equations in the sense of mean square and with probability one. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1368–1384, October, 2006.     

Asymptotic behavior of solutions to multiplicative control problems for elliptic equations

The problems of optimal multiplicative control for the Helmholtz equation and the diffusion equation are studied. The control function is included multiplicatively in a mixed-type boundary condition specified on the entire domain boundary or its part. For each of the models under study, an iterative method for determining an approximate solution is constructed and theoretically substantiated for sufficiently large values of the regularization parameter.