This book analyzes stochastic evolutionary models under the impulse ofdiffusion, as well as Markov and semi-Markov switches. Models areinvestigated under the conditions of classical and non-classical (Levyand Poisson) approximations in addition to jumping stochasticapproximations and continuous optimization procedures.Among other asymptotic properties, particular attention is given to weakconvergence, dissipativity, stability and the control of processes andtheir generators.Weak convergence of stochastic processes is usually proved by verifyingtwo conditions: the tightness of the distributions of the convergingprocesses, which ensures the existence of a converging subsequence,and the uniqueness of the weak limit. Achieving the limit can be done onthe semigroups that correspond to the converging process as well as onappropriate generators. While this provides the convergence ofgenerators, a natural question arises concerning the uniqueness of alimit semigroup.

Acronyms viiIntroduction ixChapter 1. Average Scheme and Diffusion Approximation Scheme 11.1. Stability of stochastic systems in the average scheme 11.2. Stability of stochastic systems in the diffusion approximation scheme 13Chapter 2. Levy Approximation Scheme 232.1. Differential equations with small stochastic additions in the Levy approximation scheme 232.2. Asymptotic dissipativity of stochastic processes with impulse perturbations in the Levy approximation scheme 312.3. Double merging of phase space for differential equations with small stochastic supplements under Levy approximation conditions 38Chapter 3. Asymptotical Analysis of Random Evolutionary Systems Under Poisson Approximation Conditions 513.1. Differential equations with small stochastic additions under Poisson approximation conditions 513.2. Asymptotic dissipativity of stochastic processes with impulse perturbation in the Poisson approximation scheme 583.3. Double merging of the phase space for differential equations with small stochastic supplements under Poisson approximation conditions 65Chapter 4. Stochastic Approximation Procedure 734.1. Markovenvironment 734.1.1. Jumping SAP in averaging scheme 734.1.2. Jumping SAP under diffusion approximating scheme 824.2. Semi-Markov environment 954.2.1. SAP under the averaging scheme 954.2.2. Jumping SAP in the diffusion approximation scheme 1044.3. Asymptotic normality of fluctuations of the procedure of stochastic approximation with diffusive perturbation in a Markov environment 1174.4. Asymptotic normality of SAP in a semi-Markov environment 124Chapter 5. Stochastic Optimization Procedure 1355.1. SOP in the average scheme 1355.1.1. Convergence SOP 1355.1.2. Asymptotical normality of Stochastic optimization procedure 1415.1.3. SOP with impulse perturbation 1475.2. SOP under the diffusion approximation scheme 1555.2.1. Convergence SOP 1555.2.2. Fluctuations of the stochastic optimization procedure with diffusion perturbations 1625.2.3. Fluctuation of the SOP 172Chapter 6. Combination of Approximations of Different Types 1836.1. Asymptotic properties of a stochastic diffusion process with an equilibrium point of a quality criterion 1836.2. Asymptotic properties of the impulse perturbation process with a control function under Levy approximation conditions 200References 211Index 217