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Stochastic analysis and statistical estimation for fractional and related processes

Work number - M 29 AWARDED

Presented Taras Shevchenko National University of Kyiv

Ralchenko Kostiantyn Volodymyrovych

The work is devoted to investigation of mathematical models with fractional Brownian motion and related processes and to development of statistical methods for parameter estimation in such models.

In the work we introduce a multistable subordinator, a multifractional Poisson process and mom-Euclidean generalizations of fractional Brownian and fractional Poisson fields. The properties of these processes are investigated and the limit theorems for them are obtained. We establish the conditions for existence and uniqueness of a mild solution for stochastic heat equations with fractional noise and for similar equations involving two noises, white and fractional. We prove the existence–uniqueness theorems for weak and strong solutions to stochastic differential equation with stochastic volatility. Strongly consistent drift parameter estimators for such equation are constructed. Also, we develop new methods of drift parameter estimation for a stochastic differential equation driven by fractional Brownian motion, for fractional and multifractional Ornstein–Uhlenbeck processes, and for a Gaussian process with stationary increments.

The results obtained are a contribution to the theory and statistics of stochastic processes, as well as to the theory of stochastic differential equations, both ordinary and partial. The proposed methods may be useful in the study of mathematical models of random phenomena with complex behavior, in particular those characterized by long-term or short-term dependence.

Number of publications: 52, including 2 monographs, 29 articles (21 in foreign editions), 21 abstracts. The total number of citations / h-index of work according to databases is respectively: Web of Science – 38/4; Scopus – 96/7; Google Scholar – 200/10.