Bodnarchuk S. Local properties of distributions of solutions of stochastic equations with Levy noise

Thesis is devoted to the study of local properties of distributions of solutions of stochastic differential equations with Levy noise. First group of results concerns the question for smoothness of distribution density of the solution of linear stochastic differential equations with Levy noise. In the case of one-dimensional linear stochastic differential equations with Levy noise without diffusion components and nondegenerate drift coefficient necessary and sufficient conditions for the smoothness of the distribution density of the solution of such equations are established. In the case of multivariate linear stochastic differential equations with Levy noise two groups of sufficient conditions are established. Each of these groups of conditions corresponding to various effects that may be present in the equation: the effect of "preserving smoothness" and the effect of "regularization". Second result is dedicated to studies of controllability of linear inhomogeneous integral equations. A new approach to the control of solutions of linear inhomogeneous integral equations is proposed, namely the control is performed by using time transformations. This method of control is motivated by the possibility of further application in the study of ergodic properties of Markov processes defined as the solutions to stochastic differential equations with Levy noise. The necessary and sufficient conditions for controllability of linear inhomogeneous integral equations using time transformations are obtained. The results of last section are dedicated to studies of ergodic properties of solutions of stochastic differential equations with Levy noise without diffusion components. The main theorem of section gives a lower bound for the joint part of distributions of two solutions of such equation with different initial values. The proof of this result essentially uses the method of stochastic control performed by using time transformations. The main result of the section is used to find numerical values of the constants in the exponential estimate of rate of convergence of one-dimensional distributions of a Markov process, defined as the solution of stochastic differential equations with Levy noise without diffusion component to its (unique) invariant distribution in the total variation norm.