Petranova M. Random Gaussian processes with stable correlation functions

The main objective is to find the properties of random Gaussian processes with stable correlation functions. The main topic is finding the properties and some estimates of distributions of real and complex random processes, construction of probabilistic models which approximate Gaussian process with a stable correlation function ρ2(h)=B^2*exp{-dh^2}, d>0 with a given reliability 1-α, 0<α<1 and accuracy β>0 in the spaces C([0,T]) and Lp([0,T]), p≥1, construction of a confidence interval for the parameter of the Ornstein–Uhlenbeck process and testing the hypothesis about the form of the correlation function of a centered measurable real Gaussian stationary process with a stable correlation function. Some estimates of the distribution of the supremum of a real valued Gaussian process with a stable correlation process is found; the limiting behavior of a real valued Gaussian stationary process with a stable correlation function is studied as t tends to infinity; some estimates of the norm of a real valued Gaussian random process with a stable correlation function is established in the space Lp(T); some analytic properties of Gaussian random processes with stable correlation functions are described; complex valued proper random processes are introduced and some their properties are studied; some upper bounds for the distributionss of some functionals of the absolute value of a stationary complex valued Gaussian proper processes are proved. Since the correlation function is one of the important characteristics of random processes, there are questions of evaluation and representation of this function for a random process, the construction of criteria for its identification. It is also relevant to use random Gaussian processes with stable correlation functions to solve a wide range of problems, like those for band-limited processes, as well as those in econometrics and financial mathematics. A special attention is paid to the so-called Ornstein-Uhlenbeck process as a representative of the class of Gaussian processes with a stable correlation function. The interest to the Ornstein-Uhlenbeck process has considerably grown in view of its applications in the field of finance, particularly for Vasicek Interest Rate Model. The Ornstein-Uhlenbeck process is also used for stochastic modeling of exchange rates. Recently the Ornstein-Uhlenbeck process has appeared in finance as a model of the volatility of the underlying asset price process. The results of the thesis can be conditionally divided into five parts. The first part describes, presents the properties and estimates of the distribution of real Gaussian random processes with a stable correlation function. The second part of the thesis describes, presents the properties and estimates of proper complex random process. The third part discusses the methods of representation and the main properties of Gaussian process with a stable correlation function ρ2(h)=B^2*exp{-dh^2}, d>0. Models that approximate the stationary Gaussian process with a stable correlation function ρ2(h)=B^2*exp{-dh^2}, d>0 with a given reliability, accuracy in space are constructed, and the rates of convergence of the models are found, and the corresponding theorems are stated. The next part proposes a new method for constructing a confidence interval for a parameter θ0 of the Ornstein-Uhlenbeck process as a solution of a stochastic differential equation. In the last part the problem of testing the hypothesis about the parameter of a stable correlation function is stated.