دانلود کتاب Asymptotic Analysis of Unstable Solutions of Stochastic Differential Equations

فهرست مطالب :

Preface......Page 6
Contents......Page 10
About the Authors......Page 13
Notation......Page 14
1 Introduction to Unstable Processes and Their Asymptotic Behavior......Page 15
1.1.1 Description and Motivation of the Model......Page 17
1.1.2 Asymptotic Growth and Normalizing Multipliers for the Solutions......Page 18
1.1.3 Bilayer Environment and Transition Density of the Limit Homogeneous Markov Process......Page 21
1.1.4 Comparison with the Smooth Disturbing Process......Page 23
1.1.5 Spatial Averaging of the Vibrational Type Coefficient......Page 24
1.2 Equation with the Non-unit Diffusion Coefficient......Page 25
2.1 Preliminaries......Page 29
2.2 Necessary and Sufficient Conditions for the Weak Convergence of Solutions of SDEs to a Brownian Motion in a Bilayer Environment......Page 34
2.3 Necessary and Sufficient Conditions for the Weak Convergence of Solutions of SDEs to a Process of Skew Brownian Motion Type......Page 50
2.4 Examples......Page 57
3.1 Criteria of Instability and Ergodicity for the Solutions......Page 65
3.2 Convergence of Normalized Stochastically Unstable Solutions to the Bessel Diffusion Process......Page 70
3.3 Influence of the Coefficients of the Equation on the Limit Behavior of the Solutions......Page 80
3.4 Influence of the Diffusion Coefficient on the Limit Behavior of the Solutions......Page 82
3.5 Examples......Page 86
4 Asymptotic Behavior of Integral Functionals of Stochastically Unstable Solutions......Page 90
4.1 Weak Convergence to the Functionals of a Brownian Motion in a Bilayer Environment......Page 94
4.2 Weak Convergence to the Functionals of the Bessel Diffusion Process......Page 112
4.3 Results About Weak Convergence of the Mixed Functionals......Page 128
4.4 Examples......Page 133
5.1 Preliminaries......Page 141
5.2 Theorem Concerning the Weak Compactness......Page 144
5.3 Weak Convergence to the Solutions of Itô SDEs......Page 149
5.4 Asymptotic Behavior of Integral Functionals of the Lebesgue Integral Type......Page 152
5.5 Asymptotic Behavior of Integral Functionals of Martingale Type......Page 161
5.6 Weak Convergence of Mixed Functionals......Page 167
5.7 Examples......Page 173
5.8 Auxiliary Results......Page 180
6.1 Preliminaries......Page 188
6.2 Weak Compactness and Weak Convergence of the Solutions of Itô SDEs......Page 191
6.3 Asymptotic Behavior of Integral Functionals of the Lebesgue Integral Type......Page 195
6.4 Weak Convergence of Martingale Type Functional and of Mixed Functional......Page 200
6.5 Examples......Page 201
6.6 Auxiliary Results......Page 203
A.1.1 Basic Facts Regarding Stochastic Processes......Page 207
A.1.1.3 Wiener Process as an Example of a Gaussian Process......Page 209
A.1.2 Notion of Stochastic Basis with Filtration......Page 210
A.1.3 Notion of (Sub-, Super-) Martingale. Elementary Properties. Square-Integrable Martingales. Quadratic Variations and Quadratic Characteristics......Page 211
A.1.4 Markov Moments and Stopping Times......Page 213
A.1.5 Construction of Stochastic Integral w.r.t. a Wiener Process and a Square-Integrable Continuous Martingale......Page 214
A.1.6 Generalized Itô Formula......Page 217
A.1.7 Skorokhod's Representation Theorem and Prokhorov's Theorem......Page 219
A.2 Convergence of Stochastic Integrals and Some Properties of Solutions of SDEs......Page 220
A.3 Brownian Motion in a Bilayer Environment......Page 235
A.4 Functions Regularly Varying at Infinity......Page 244
References......Page 246