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Information × Registration Number 0217U001377, 0116U002530 , R & D reports Title Investigation and statistical analysis of the asymptotic behavior of complex stochastic nonhomogeneous dynamic systems popup.stage_title Head Mishura Yuliya Stepanivna, Registration Date 08-02-2017 Organization Taras Shevchenko Kiev University popup.description2 The objects of investigation are random processes and fields, financial markets, nonstandard stochastic differential equations and integral metrics. The aims of the work are development of theory of mixed fractional and multifractional processes, establishing of functional limit theorems for financial markets, approximation of paths of random processes with given accuracy and reliability, development of methods for optimal estimation of functionals of unknown values of stochastic processes and random fields, development of new methods for nonparametric statistical analysis of mixtures. The methods of investigation are analytic methods of probability theory and mathematical statistics, methods of stochastic analysis, methods of stochastic modelling and new statistical methods. Main results of investigation obtained in the work: We establish asymptotic growth bounds for multifractional Gaussian processes. We prove the strong consistency of maximum likelihood estimates for the drift parameter in the multifractional nonergodic Ornstein-Uhlenbeck model. We construct a testing criterion for the hypothesis of the drift coefficient sign in the fractional Ornstein-Uhlenbeck process. For the stochastic heat equation with stable measure, we prove the existence of a solution and convergence of approximations which are defined by truncating the LePage series. We find accuracy and reliability of the Whittaker-Kotel'nikov-Shannon approximation for stationary phi-sub-Gaussian random processes with bounded spectrum in L_p(T) and C(T). For a function approximation problem in integral metrics, we prove that the rate of the best monotone (or convex) polynomial approximation in the integral norm with Jacobi weight is bounded by the second (third) module of smoothness with the same weight, and this rate cannot be bounded by the third (forth) module. We study estimation (interpolation, extrapolation) problems for functionals of unknown values of harmonized stochastic processes and stationary processes on the basis of observations with missing values under conditions of spectral certainty and spectral uncertainty; we study similar problems for random fields. For an arbitrage-free market with stocks of one type, we study conditions when successive stock prices form a comonotone vector. For an arbitrage-free market with stocks of one type, we describe consequences of the assumption of the vector comonotonicity. For models with measurement errors, we estimate parameters of two straight lines on the basis of disturbed observations of points on these lines and study the following estimates: an estimate projection using the method of estimator adjustment, a maximum likelihood estimate in the parametric model and an asymptotically optimal estimate in some class of estimates using the method of moments. We obtain a modification of the Kaplan-Meier estimator for the distribution of components of a mixture with variable concentrations on the basis of censored data and investigate its asymptotics when the sample size increases. We obtain asymptotic normality conditions for spectral functionals that are built on the basis of weighted observations of homogeneous Gaussian random fields with strong dependence. We get bounds for the correlation function of a Gaussian stationary process. We propose a new testing criterion for the hypothesis of a view of correlation function of Gaussian stationary random process with unknown mean. We prove functional limit theorems for financial markets and obtain exact and approximate option prices on financial markets with stochastic volatility. We study practical approaches to the ruin probability estimation in a risk model with additional funds. The results of the work have been introduced into the teaching process at the Faculty of Mechanics and Mathematics of Taras Shevchenko National University of Kyiv. MULTIFRACTIONAL PROCESS, GAUSSIAN PROCESS, FINANCIAL MARKET, ACCURACY AND RELIABILITY OF AN APPROXIMATION, INTEGRAL NORM, ESTIMATION PROBLEM, MIXTURE WITH VARIABLE CONCENTRATIONS, COMONOTONE VECTOR, SPECTRAL FUNCTIONAL, HYPOTHESIS TESTING, CORRELATION FUNCTION. Product Description popup.authors Зубченко В.П. Зутула Д.В. Козаченко Ю.В. Кукуш О.Г. Мішура Ю.С. Майборода Р.Є. Моклячук М.П. Рагуліна О.Ю. Сахно Л.М. Шевченко Г.М. Шевчук І.О. Шкляр С.В. popup.nrat_date 2020-04-02 Close