Avetisian D. Parameter estimation in stochastic partial differential equations with fractional noises

The thesis is devoted to fractional stochastic partial differential equations driven by Wiener process, fractional Brownian motion or their linear combination. The main goal of the research is the development of statistical methods for simultaneous noise parameters estimation based on discrete observations of the solutions. Also asymptotic properties of estimators have been investigated. Special attention is given to the properties of solutions such as stationarity and ergodicity since they are crucial for the construction and further investigation of statistical estimators. The rapid development of the theory of partial differential equations is in progress for the last four decades. This theory combines the elements of stochastic partial differential equations theory and stochastic analysis. It can be applied in many scientific areas such as physics, biology, geophysics and finances. Such equations are used to model the diffusion processes, phase transitions and material properties. They also can be applied in finance for option pricing, risk management and stochastic volatility modeling. The investigation of the stochastic partial differential equations properties and development of the statistical methods for them are important problems for modern studies as they are playing a crucial role in many scientific fields. The special attention in this thesis is devoted to stochastic partial differential equations driven by fractional Brownian noise. Such type of equations can be used to describe the processes with long-term and short-term dependencies which are important for physical systems, radio-electronic devices, computer networks and financial markets. In addition, this research covers more complex models with combination of white and fractional Brownian noises. This allows us, in particular, to model financial market processes more accurately when different sources of randomness exist.