# Volume 47, № 7, 1995

### On the 70th birthday of Vladimir Semenovich Korolyuk

Ukr. Mat. Zh. - 1995. - 47, № 7. - pp. 867-868

### To the problem of canonical factorization for Markov additive processes

Ukr. Mat. Zh. - 1995. - 47, № 7. - pp. 869–875

We extend the well-known results on canonical factorization for Markov additive processes with a finite Markov chain to the case where this chain is countable. We also formulate some corollaries of these results.

### On the properties of an empirical correlogram of a Gaussian process with square integrable spectral density

Ukr. Mat. Zh. - 1995. - 47, № 7. - pp. 876–889

We study properties of an empirical correlogram of a centered stationary Gaussian process. We prove that if the spectral density of the process is square integrable, then there is a normalization effect for the correlogram and integral functionals of it.

### Bounded solutions of one class of nonlinear operator difference equations

Dorogovtsev A. Ya., Gorodnii M. F.

Ukr. Mat. Zh. - 1995. - 47, № 7. - pp. 890–896

Sufficient conditions for existence of bounded solutions of operator difference equations with quadratic nonlinearity are given.

### On crossing of a level by processes defined by sums of a random number of terms

Ukr. Mat. Zh. - 1995. - 47, № 7. - pp. 897–914

We study the joint distribution of boundary functionals related to the crossing of a positive (negative) level by a process consisting of a homogeneous Poisson process and a process defined by sums of a random number of continuously distributed terms.

### Nonlocal boundary-value problem for parabolic equations with variable coefficients

Ptashnik B. I., Zadorozhna N. M.

Ukr. Mat. Zh. - 1995. - 47, № 7. - pp. 915–921

We study the boundary-value problem for Petrovskii parabolic equations of arbitrary order with variable coefficients with conditions nonlocal in time. We establish conditions for the existence and uniqueness of a classical solution of this problem and prove metric theorems on lower bounds of small denominators appearing in the construction of a solution of the problem.

### Normal approximation of random permanents

Borovskikh Yu. V., Korolyuk V. S.

Ukr. Mat. Zh. - 1995. - 47, № 7. - pp. 922–927

For random permanents, we obtain an estimate of the rate of convergence in the central limit theorem.

### Diffusion approximation of stochastic Markov models with persistent regression

Ukr. Mat. Zh. - 1995. - 47, № 7. - pp. 928–935

Sequences of sums of identically distributed random variables forming a homogeneous Markov chain are approximated by a time-discrete autoregression process of Ornstein-Uhlenbeck type.

### Integral approximation of stochastic differential equations with anticipating initial conditions

Ukr. Mat. Zh. - 1995. - 47, № 7. - pp. 936–945

We give a sequence of stochastic integro-differential equations that approximates a stochastic differential equation with an anticipating initial condition and localized Skorokhod stochastic integral. A sequence of solutions of these equations is obtained. The convergence of this sequence to a certain process implies that this process is a solution (generally speaking, local) of the original equation.

### Estimates of intense noise for inhomogeneous diffusion processes

Ukr. Mat. Zh. - 1995. - 47, № 7. - pp. 946–951

Nonparametric estimates of noise intensity*g(t)* are obtained. These estimates are constructed for data*x*(*t*) defined by the equation*dx(t)=f(x(t),t)dt+g(t)dw(t)*. The validity of the estimates is proved.

### Two-parameter Lévy processes: ItÔ formula, semigroups, and generators

Ukr. Mat. Zh. - 1995. - 47, № 7. - pp. 952–961

We consider random Lévy fields, i.e., stationary fields continuous in probability and having independent increments. We prove that the trajectories of such fields have at most one jump on every line parallel to the axes. We derive an expression for the ItÔ change of variables for Lévy fields. We also consider semigroups generated by Lévy fields and their generators.

### Robust interpolation of random fields homogeneous in time and isotropic on a sphere, which are observed with noise

Ukr. Mat. Zh. - 1995. - 47, № 7. - pp. 962–970

We study the problem of optimal linear estimation of the functional $$A_N \xi = \sum\limits_{k = 0}^{\rm N} {\int\limits_{S_n } {a(k,x)\xi (k,x)m_n (dx),} }$$ , which depends on unknown values of a random field ξ(*k, x*),*k*∃Z,*x∃S* _{n} homogeneous in time and isotropic on a sphere*S* _{n}, by observations of the field ξ(*k,x*)+η(*k,x*) with k∃ Z{0, 1, ...,*N*},*x*∃S_{n} (here, η (*k, x*) is a random field uncorrelated with ξ(*k, x*), homogeneous in time, and isotropic on a sphere S_{n}). We obtain formulas for calculation of the mean square error and spectral characteristic of the optimal estimate of the functional*A* _{N}ξ. The least favorable spectral densities and minimax (robust) spectral characteristics are found for optimal estimates of the functional*A* _{N}ξ.

### Superfractality of the set of numbers having no frequency of*n*-adic digits, and fractal probability distributions

Pratsiovytyi M. V., Torbin H. M.

Ukr. Mat. Zh. - 1995. - 47, № 7. - pp. 971–975

We study the fractal properties (we find the Hausdorff-Bezikovich dimension and Hausdorff measure) of the spectrum of a random variable with independent*n*-adic (*n*≥2,*n* ∃*N* digits, the infinite set of which is fixed. We prove that the set of numbers of the segment [0, 1] that have no frequency of at least one*n*-adic digit is superfractal.

### Hedging of options under mean-square criterion and semi-Markov volatility

Ukr. Mat. Zh. - 1995. - 47, № 7. - pp. 976–983

We consider a problem of hedging of the European call option for a model in which the appreciation rate and volatility are functions of a semi-Markov process. In such a model, the market is incomplete.

### Estimates in the Rényi theorem for differently distributed terms

Ukr. Mat. Zh. - 1995. - 47, № 7. - pp. 984–989

We obtain estimates for the rate of convergence of the distribution function of a sum of a geometric number of differently distributed random variables to a function of a special kind in the case where the parameter of the geometric distribution tends to zero. We also consider the problem of convergence of inhomogeneous thinning flows, which is closely related to the geometric summation.

### Mean-square asymptotic stability of solutions of systems of stochastic differential equations with random operators

Yasinskaya L. I., Yasinsky V. K., Yurchenko I. V.

Ukr. Mat. Zh. - 1995. - 47, № 7. - pp. 990–1001

We obtain conditions of asymptotic behavior of trivial solutions of systems of stochastic differential equations with random operators.

### Random permanents of mixed multisampling matrices

Ukr. Mat. Zh. - 1995. - 47, № 7. - pp. 1002–1005

We consider the weak convergence of random permanents under certain conditions.

### The limit theorem for the maximum ot $C$-valued Gaussian random variables

Ukr. Mat. Zh. - 1995. - 47, № 7. - pp. 1006-1008

The well known asymptotic equality for the maximum of real Gaussian random variables is generalized to the case of random variables with the values taken in the space $C$.