Asymptotic and Analytic Methods in Stochastic Evolutionary Symptoms

Dmitri Koroliouk is a Doctor of Sciences, Professor at the National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, and leading researcher at the Institute of Mathematics, and at the Institute of Telecommunications and Global Information Space of the National Academy of Sciences of Ukraine. He is also Head of the Digital Innovation Laboratory at UNESCO Interdisciplinary Chair in Biotechnology and Bioethics, at the University of Rome Tor Vergata, Italy. Igor Samoilenko is a Doctor of Sciences, Professor at the Taras Shevchenko National University of Kyiv, and Professor at the Institute for Applied System Analysis, part of the National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”.

• Preface ixIntroduction xiChapter 1 Multidimensional Models of Kac Type 11.1. Definitions and basic properties 11.2. Moments of evolutionary process 81.3. Systems of Kolmogorov equations 171.4. Evolutionary operator and theorem about weak convergence to the measure of the Wiener process 23Chapter 2 Symmetry of Markov Random Evolutionary Processes in Rn 292.1. Symmetrization: definition and properties 292.2. Examples of symmetric distributions in Rn and distributions on n + 1-hedra322.2.1. Symmetric distributions 322.2.2. Distributions on n + 1-hedra 35Chapter 3 Hyperparabolic Equations, Integral Equation and Distribution for Markov Random Evolutionary Processes 393.1. Hyperparabolic equations and methods of solving Cauchy problems 393.2. Analytical solution of a hyperparabolic equation with real-analytic initial conditions 463.3. Integral representation of the hyperparabolic equation 573.4. Distribution function of evolutionary process 67Chapter 4 Fading Markov Random Evolutionary Process 774.1. Definition of fading Markov random evolutionary process, its moments and limit distribution 774.2. Integral equation for a function from the fading random evolutionary process 894.3. Equations in partial derivatives for a function of the fading random evolutionary process 93Chapter 5 Two Models of the Evolutionary Process 995.1. Evolution on a complex plane 995.2. Evolution with infinitely many directions 1095.2.1. Symmetric case 1105.2.2. Non-symmetric case 119Chapter 6 Diffusion Process with Evolution and Its Parameter Estimation 1256.1. Asymptotic diffusion environment 1256.2. Approximation of a discrete Markov process in asymptotic diffusion environment 1276.3. Parameter estimation of the limit process 132Chapter 7 Filtration of Stationary Gaussian Statistical Experiments 1357.1. Introduction 1357.2. Stochastic difference equation of the process of filtration 1377.3. Coefficient of filtration 1387.4. Equation of optimal filtration 1397.5. Characterization of a filtered signal 141Chapter 8 Adapted Statistical Experiments with Random Change of Time 1438.1. Introduction 1438.2. Statistical experiments and evolutionary processes 1448.3. Stochastic dynamics of statistical experiments 1458.4. Adapted statistical experiments in series scheme 1478.5. Convergence of the adapted statistical experiments 1498.6. Scaling parameter estimation 1548.7. Statistical estimations of the renewal intensity parameter 1558.7.1. Poisson’s renewal process with parameter q =2 1568.7.2. Stationary renewal process with delay, determined by the initial distribution function of the limit over jumps 1568.7.3. Renewal processes with arbitrarily distributed renewal intervals 157Chapter 9 Filtering of Stationary Gaussian Statistical Experiments 1599.1. Stationary statistical experiments 1599.2. Filtering of discrete Markov diffusion 1619.3. The filtering error 1649.4. The filtering empirical estimation 166Chapter 10 Asymptotic Large Deviations for Markov Random Evolutionary Process 17110.1. Asymptotic large deviations 17110.2. Asymptotically stopped Markov random evolutionary process 19110.3. Explicit representation for the normalizing function 206Chapter 11 Asymptotic Large Deviations for Semi-Markov Random Evolutionary Processes 20911.1. Recurrent semi-Markov random evolutionary processes 20911.2. Asymptotic large deviations 212Chapter 12 Heuristic Principles of Phase Merging in Reliability Analysis 22112.1. The duplicated renewal system 22112.2. The duplicated renewal system in the series scheme 22212.3. Heuristic principles of the phase merging 22312.4. The duplicated renewal system without failure 225References 227Index 233