Volume 52, № 9, 2000
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1155-1157
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1158-1165
We construct a process of Brownian motion on the Siegel disk of infinite dimension.
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1166-1175
We investigate necessary and sufficient conditions for the almost-sure boundedness of normalized solutions of linear stochastic differential equations in $R^d$ their almost-sure convergence to zero. We establish an analog of the bounded law of iterated logarithm.
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1176-1193
We study the properties of the noise (in the Tsirelson sense) that is generated by the solutions of the well-known Tanaka equation.
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1194-1204
We consider the structure of orthogonal polynomials in the space L 2(B, μ) for a probability measure μ on a Banach space B. These polynomials are described in terms of Hilbert–Schmidt kernels on the space of square-integrable linear functionals. We study the properties of functionals of this sort. Certain probability measures are regarded as generalized functionals on the space (B, μ).
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1205-1207
We show that the analysis of the global behavior of a process with independent increments in terms of the existence of the stationary distribution of the corresponding storage process leads to results that differ from the classical ones.
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1208-1218
We solve the problem of extrapolation of an analytic function of a certain class in the case where its values are observed in a white noise whose intensity is not high.
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1219-1225
We consider a queuing system (≤ λ)/G/m, where the symbol (≤ λ) means that, independently of prehistory, the probability of arrival of a call during the time interval dtdoes not exceed λdt. The case where the queue length first attains the level r≥ m+ 1 during a busy period is called the refusal of the system. We determine a bound for the intensity μ1(t) of the flow of homogeneous events associated with the monotone refusals of the system, namely, μ1(t) = O(λ r+ 1α1 m− 1α r− m+ 1), where α k is the kth moment of the service-time distribution.
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1226-1250
We investigate the properties of the image of a differentiable measure on an infinitely-dimensional Banach space under nonlinear transformations of the space. We prove a general result concerning the absolute continuity of this image with respect to the initial measure and obtain a formula for density similar to the Ramer–Kusuoka formula for the transformations of the Gaussian measure. We prove the absolute continuity of the image for classes of transformations that possess additional structural properties, namely, for adapted and monotone transformations, as well as for transformations generated by a differential flow. The latter are used for the realization of the method of characteristics for the solution of infinite-dimensional first-order partial differential equations and linear equations with an extended stochastic integral with respect to the given measure.
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1251-1256
For inhomogeneous systems of Itô stochastic differential equations, we introduce the notion of local invariance of surfaces and the notion of local first integral. We obtain results that give the general possibility of finding invariant surfaces and functionally independent first integrals of stochastic differential equations.
Properties of the Likelihood Ratio for Counting Processes in the Problem of Estimation of Unknown Parameters
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1257-1268
We obtain an asymptotic decomposition of the logarithm of the likelihood ratio for counting processes in the case of similar hypotheses. We establish the properties of the normalized likelihood ratio in the problem of estimation of an unknown parameter.
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1269-1271
We obtain some results concerning the upper limit of a random sequence and the law of the iterated logarithm for sums of independent random variables.
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1272-1282
We consider a multidimensional Wiener process with a semipermeable membrane located on a given hyperplane. The paths of this process are the solutions of a stochastic differential equation, which can be regarded as a generalization of the well-known Skorokhod equation for a diffusion process in a bounded domain with boundary conditions on the boundary. We randomly change the time in this process by using an additive functional of the local-time type. As a result, we obtain a probabilistic representation for solutions of one problem of mathematical physics.
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1283-1293
We improve the known upper and lower bounds for the probability of the fact that exactly k ievents should occur in a group consisting of n ievents simultaneously for all i= 1, 2, ..., d.
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1294-1303
We prove that the amplitudes and the phases of eigenoscillations of a linear oscillating system perturbed by either a fast Markov process or a small Wiener process can be described asymptotically as a diffusion process whose generator is calculated.
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1304-1309
We find the exact distribution of an arbitrary remainder of an infinite sum of overlapping products of a sequence of independent Bernoulli random variables.