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

Yaroslav Chabanyuk is Senior Research Officer at Lublin University of Technology, Poland, and at Ivan Franko National University of Lviv, Ukraine. His work specializes in system analysis and properties of random processes with Markov and semi-Markov switching.Anatolii Nikitin is Senior Research Officer at Taras Shevchenko National University of Kyiv, Ukraine, and Professor at the Jan Kochanowski University (JKU) in Kielce, Poland. His work specializes in system analysis and asymptotic properties of stochastic differential equations.Uliana Khimka is Professor at Ivan Franko National University of Lviv. She is the author of more than 100 scientific works and, in 2013, she defended her PhD thesis, Continuous Stochastic Optimizations Procedure in the Environment with Markov Switching.

• 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