Nikitin A. Analysis of asymptotic properties of stochastic differential equations with the help of approximation schemes.

Analyzing the state of the art concerning asymptotic properties of stochastic evolution models reveals that a complete theory is still to be worked out. Well understood are the models which are given by stochastic differential equations with Markov switchings and impulse or continuous-type perturbations in the classical schemes of averaging or diffusion approximation. Also, the asymptotic behavior was investigated of impulse processes with Markov switchings under the conditions of Levy or Poisson approximation. Thus, it seems natural to develop a theory of evolution equations with Markov switchings and random perturbations in nonclassical approximation schemes. The present thesis is concerned with the system analysis of asymptotic properties of evolution models which are given by stochastic differential equations. In particular, we consider stochastic differential equations with Markov switchings and impulse perturbations under the conditions of Levy and Poisson approximation, controlled stochastic differential equations with Markov switchings and diffusion perturbations (assuming uniqueness of the equilibrium point for the quality criterion for which the stochastic approximation procedure is given), and some limit models given by either Ito-Skorokhod stochastic differential equations or stochastic differential-difference equations with delays. The asymptotic properties of the aforementioned stochastic models are systematically investigated from different viewpoints. In particular, we construct generators of the limit processes and the limit control with the point of equilibrium of the quality criterion function, prove asymptotic dissipativity and asymptotic stability in the mean square and exploit the double merging of the phase space. While doing so we use several approximation schemes, namely, Levy approximation, Poisson approximation and stochastic approximation. The results obtained in the present thesis may be divided into two parts. In the first part we consider some prelimit evolution models with a small normalization parameter. We find the form of the limit generators for the impulse or diffusion processes and the dynamical system in the schemes of the Poisson approximation, the Levy approximation, and the stochastic approximation. Further, in this part we provide conditions which ensure weak convergence of a controlled evolution model with Markov switching and diffusion perturbation (assuming uniqueness of the equilibrium point for the quality criterion for which the stochastic approximation procedure is given). Also, we show that the transfer process, properly normalized, converges weakly to an Ornstein–Uhlenbeck process. It is important that in this part of the thesis the asymptotic behavior of the limit process is concluded with the help of the analysis of parameters of the prelimit system. Weak convergence of stochastic processes is usually proved by checking the two conditions: (a) tightness of the distributions of the converging processes which ensures the existence of a converging subsequence and (b) uniqueness of the weak limit. The passage to the limit can be done on the semigroups which correspond to the converging processes as well as on appropriate generators. While proving convergence of generators a natural question arises concerning the uniqueness of a limit semigroup. It can be answered by representing the process in focus as a unique solution to a martingale problem which is formulated with the help of the limit generator.