Tymoshenko O. The asymptotic behavior of solutions of stochastic differential equations

The thesis is devoted to the study of asymptotic behaviour of stochastic differential equation solution. Exact order of growth of solution of equation peturbed by Wiener process with time depended coefficient of drift and diffusion are presented in the work. Solution of equation are supposed to tend to infinity as time growth unboundedly to infinity. Notion of psi-asymptotic equivalence of one dimentional autonomous stochastic differential equation solution and ordinary differential equation solution are proposed. It helps to compare behaviour of solutions in the case of unboundedness of difference between of them. Nessesary and sufficient conditions of psi-asymptotic equivalence of autonomous stochastic differential equation solutions and ordinary differential equation solutions are obtained.Moreover, sufficient conditions of unboundedness of nonautonomous stochastic differential equation solution are stated.