Asymptotic Analyses for Complex Evolutionary Systems with Markov and Semi-Markov Switching Using Approximation Schemes

This book analyzes stochastic evolutionary models under the impulse of diffusion, as well as Markov and semi-Markov switches. Models are investigated under the conditions of classical and non-classical (Levy and Poisson) approximations in addition to jumping stochastic approximations and continuous optimization procedures. Among other asymptotic properties, particular attention is given to weak convergence, dissipativity, stability and the control of processes and their generators. Weak convergence of stochastic processes is usually proved by verifying two conditions: the tightness of the distributions of the converging processes, which ensures the existence of a converging subsequence, and the uniqueness of the weak limit. Achieving the limit can be done on the semigroups that correspond to the converging process as well as on appropriate generators. While this provides the convergence of generators, a natural question arises concerning the uniqueness of a limit semigroup.

Table of Contents:
Acronyms vii Introduction ix Chapter 1. Average Scheme and Diffusion Approximation Scheme 1 1.1. Stability of stochastic systems in the average scheme 1 1.2. Stability of stochastic systems in the diffusion approximation scheme 13 Chapter 2. Levy Approximation Scheme 23 2.1. Differential equations with small stochastic additions in the Levy approximation scheme 23 2.2. Asymptotic dissipativity of stochastic processes with impulse perturbations in the Levy  approximation scheme 31 2.3. Double merging of phase space for differential equations with small stochastic supplements under Levy approximation conditions 38 Chapter 3. Asymptotical Analysis of Random Evolutionary Systems Under Poisson Approximation Conditions 51 3.1. Differential equations with small stochastic additions under Poisson approximation conditions 51 3.2. Asymptotic dissipativity of stochastic processes with impulse perturbation in the Poisson approximation scheme 58 3.3. Double merging of the phase space for differential equations with small stochastic supplements under Poisson approximation conditions 65 Chapter 4. Stochastic Approximation Procedure 73 4.1. Markovenvironment 73 4.1.1. Jumping SAP in averaging scheme 73 4.1.2. Jumping SAP under diffusion approximating scheme 82 4.2. Semi-Markov environment 95 4.2.1. SAP under the averaging scheme 95 4.2.2. Jumping SAP in the diffusion approximation scheme 104 4.3. Asymptotic normality of fluctuations of the procedure of stochastic approximation with diffusive perturbation in a Markov environment 117 4.4. Asymptotic normality of SAP in a semi-Markov environment 124 Chapter 5. Stochastic Optimization Procedure 135 5.1. SOP in the average scheme 135 5.1.1. Convergence SOP 135 5.1.2. Asymptotical normality of Stochastic optimization procedure 141 5.1.3. SOP with impulse perturbation 147 5.2. SOP under the diffusion approximation scheme 155 5.2.1. Convergence SOP 155 5.2.2. Fluctuations of the stochastic optimization procedure with diffusion perturbations 162 5.2.3. Fluctuation of the SOP 172 Chapter 6. Combination of Approximations of Different Types 183 6.1. Asymptotic properties of a stochastic diffusion process with an equilibrium point of a quality criterion 183 6.2. Asymptotic properties of the impulse perturbation process with a control function under Levy approximation conditions 200 References 211 Index 217