Vasylyk O. Generalization of φ-sub-Gaussian stochastic processes and their application

The thesis is devoted to the study of properties of φ-sub-Gaussian stochastic processes and stochastic processes from the class V (φ; ), as well as to their application. The class of φ-sub-Gaussian stochastic pro- cesses generalizes classes of Gaussian and sub-Gaussian random processes and is very interesting from the point of view of investigation and simu- lation of real stochastic processes, which appear, for example, in queuing systems and nancial mathematics. In particular, among such processes there are processes of fractional Brownian motion. New estimates for dis- tribution of suprema, conditions for sample continuity with probability one and estimates for distribution of increments of stochastic processes from the class V (φ; ) are obtained. There are studied φ-sub-Gaussian random processes from the point of view of Lipschitz continuity, and esti- mates of distributions of norms of such processes are obtained. There are found modula of continuity and conditions, under which φ-sub-Gaussian stochastic processes belong to Lipschitz spaces. The results obtained are applied to weakly self-similar random processes with stationary incre- ments. Wavelet expansions of random processes from the class V (φ; ) are investigated. Conditions for uniform convergence with probability one of wavelet expansions of such processes are derived. Basing on the results obtained for φ-sub-Gaussian random processes, there are constructed algorithms for simulation of such processes with given reliability and accuracy in the spaces C([0; T]) and Lp([0; T]). Two approaches are used: simulation on the base of series expansion and simu- lation on the base of spectral representation of considered processes. The results obtained are applied to simulation of fractional Brownian motion. In the thesis, there are also studied properties of some strictly φ-sub- Gaussian random eld, which is generated by φ-sub-Gaussian random process. Estimates for distribution of suprema of such a random eld are obtained and the rate of growth of the eld on unbounded domain is estimated. The notion of a strictly φ-sub-Gaussian quasi shot noise process, gen- erated by φ-sub-Gaussian random processes and some response function, is introduced and properties of such processes are studied. The shot noise processes are mathematical models of various phenomena and have been studied since the beginning of the XX-th century. Now, the shot noise processes are applied not only in physics, but also in insurance, nancial mathematics, telecommunication networks theory. In the classical shot noise model, it was supposed that impulses arrive to some systems in ac- cordance with a Poisson process, but over time many interesting modica- tions of the model have been developed. In the thesis, there are obtained conditions for sample continuity of a strictly φ-sub-Gaussian quasi shot noise process dened on a compact set, as well as conditions under which such processes belong to some weighted spaces of continuous functions in the case, when the processes are dened on the real line. Estimates for distributions of suprema of the strictly φ-sub-Gaussian quasi shot noise processes are derived.