Krenevych A. Asymptotic investigation of stochastic differential systems in finite-dimensional and Hilbert spaces

The thesis is dedicated to investigation of the asymptotic behaviour of solutions to the stochastic differential Ito's equations in finite-dimensional and Hilbert spaces. The exponential dichotomy conditions of a linear homogeneous stochastic system are obtained. These conditions are determined in terms of bounded solutions and quadratic forms. The conditions of the asymptotic equivalence in the mean square and with probability one are derived for the stochastic differential systems in finite-dimensional and Hilbert spaces. The existence and uniqueness theorem for solutions of the stochastic systems not resolved with respect to "derivative" in Hilbert space are proved.