Random Evolutionary Systems - Asymptotic Properties and Large Deviations

Cover
Half-Title Page
Dedication
Title Page
Copyright Page
Contents
Preface
Introduction
Chapter 1. Basic Tools for Asymptotic Analysis
1.1. Basic concepts of operator asymptotic analysis
1.1.1. Reducibly invertible and potential operators
1.1.2. Singular perturbation problem
1.1.3. Markov process
1.1.4. Semi-Markov process
1.1.5. Phase merging
1.1.6. Processes with independent increments
1.1.7. Poisson approximation scheme for the processes with independent increments
1.1.8. Random evolutionary systems with locally independent increments
1.1.9. Impulse recurrent processes
1.1.10. Random evolutionary systems and impulsive processes in the Poisson approximation scheme
1.2. Nonlinear exponential generator of large deviations, Nisio semigroup and control problem
1.2.1. Martingale characterization and Brick’s formula
1.2.2. Nonlinear exponential generator of large deviations
1.2.3. Large deviations problem in the Poisson approximation scheme
1.2.4. Nisio semigroup and control problem
1.3. Compactness and comparison principle
1.3.1. Compactness and exponential compactness
1.3.2. Comparison principle
Chapter 2. Weak Convergence in Poisson and Lévy Approximation Schemes
2.1. Random evolutionary systems with locally independent increments
2.1.1. Markov switching
2.1.2. Semi-Markov switching
2.1.3. Unbounded jump measures
2.2. Impulsive recurrent process
2.2.1. Markov switching
2.2.2. Semi-Markov switching
Chapter 3. Large Deviations in the Scheme of Asymptotically Small Diffusion
3.1. Statement of the problem
3.2. Processes with locally independent increments
3.3. Random evolutionary systems in the scheme of ergodic phase merging
3.3.1. Large deviations problem under the balance condition (total)
3.3.2. Large deviations problem under the (local) balance condition
3.4. Markov integral functional
Chapter 4. Large Deviations of Systems in Poisson and Lévy Approximation Schemes
4.1. Random evolutionary systems with independent increments
4.1.1. Poisson approximation scheme
4.1.2. Lévy approximation scheme
4.2. Impulsive processes
4.2.1. Poisson approximation scheme
4.2.2. Lévy approximation scheme
Chapter 5. Large Deviations of Systems in the Scheme of Splitting and Double Merging
5.1. Small diffusion scheme
5.1.1. Large deviations under the local balance (LB) condition
5.1.2. Large deviations under the total balance (TB) condition
5.2. Poisson approximation scheme
5.3. Lévy approximation scheme
Chapter 6. Difference Diffusion Models with Equilibrium
6.1. Statistical experiments with linear persistent regression
6.1.1. Persistent regression
6.1.2. Equilibrium state
6.1.3. Convergence of statistical experiments to the equilibrium state
6.1.4. Approximation of statistical experiments by a normal process of autoregression
6.2. Exponential statistical experiments
6.2.1. Steady regime for exponential statistical experiments
6.2.2. Approximation of an exponential statistical experiment by a normal process of autoregression
6.3. Statistical experiments with nonlinear persistent regression
6.3.1. Equilibrium state
6.3.2. Approximation by a normal process of autoregression
6.3.3. Proof of Theorem 6.6
6.4. Difference diffusion models with two equilibriums
6.4.1. The principle of “stimulation and restraint”
6.4.2. Difference evolutionary model
6.4.3. Interpretation of zones of influence of equilibriums π± in the economic space
6.4.4. Difference stochastic model
6.4.5. Classification of equilibriums of the stochastic model statistical experiment
6.4.6. Approximation of the stochastic component
6.4.7. Approximation of statistical experiment in discrete–continuous time
6.5. Multivariate statistical experiments with persistent nonlinear regression and equilibrium
6.5.1. Basic definitions and assumptions
6.5.2. Equilibrium and transformation of regression function
6.5.3. Examples
6.5.4. The state of equilibrium of multivariate statistical experiments
6.5.5. Stochastic approximation of multivariate statistical experiments
6.6. Multivariant Wright–Fisher model
6.6.1. Regression function of increments
6.6.2. Equilibrium state
6.7. Binary evolutionary process
Chapter 7. Random Evolutionary Systems in Discrete–Continuous Time
7.1. Discrete Markov evolutions in an asymptotic diffusion environment
7.1.1. Asymptotic diffusion perturbation
7.2. Discrete Markov process with asymptotically small diffusion
7.2.1. Asymptotically small diffusion
7.2.2. Exponential generator of the discrete Markov process
7.2.3. Rate functional of the discrete Markov process
7.3. The problem of discrete Markov random evolution leaving an interval
Chapter 8. Diffusion Approximation of Random Evolutions in Random Media
8.1. Binary discrete Markov evolutions
8.1.1. Discrete Markov evolutions
8.1.2. Justification of diffusion approximation
8.2. Multivariate random evolutionary systems in discrete–continuous time
8.2.1. Evolutionary process in discrete–continuous time
8.2.2. Difference stochastic equation
8.2.3. Diffusion approximation of random evolutionary systems in discrete–continuous time
8.3. Discrete random evolutionary systems in a Markov random environment
8.3.1. Discrete and continuous Markov random environments
8.4. Random evolutionary systems in a balanced Markov random environment
8.4.1. Basic assumptions
8.5. Adapted random evolutionary systems
8.5.1. Bernoulli approximation of the discrete Markov diffusion stochastic component
8.5.2. Adapted random evolutionary systems
8.5.3. Adapted random evolutionary systems in a series scheme
References
Index
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