Applied Stochastic Analysis (Graduate Studies in Mathematics, 199)

This is a textbook for advanced undergraduate students and beginning graduate students in applied mathematics. It presents the basic mathematical foundations of stochastic analysis (probability theory and stochastic processes) as well as some important practical tools and applications (e.g., the connection with differential equations, numerical methods, path integrals, random fields, statistical physics, chemical kinetics, and rare events). The book strikes a nice balance between mathematical formalism and intuitive arguments, a style that is most suited for applied mathematicians. Readers can learn both the rigorous treatment of stochastic analysis as well as practical applications in modeling and simulation. Numerous exercises nicely supplement the main exposition. Cover......Page 1 Title page......Page 4 Introduction to the Series......Page 14 Preface......Page 18 Notation......Page 20 Part 1 . Fundamentals......Page 24 1.1. Elementary Examples......Page 26 1.2. Probability Space......Page 28 1.3. Conditional Probability......Page 29 1.4. Discrete Distributions......Page 30 1.5. Continuous Distributions......Page 31 1.6. Independence......Page 35 1.7. Conditional Expectation......Page 37 1.8. Notions of Convergence......Page 39 1.9. Characteristic Function......Page 40 1.10. Generating Function and Cumulants......Page 42 1.11. The Borel-Cantelli Lemma......Page 44 Exercises......Page 47 Notes......Page 50 2.1. The Law of Large Numbers......Page 52 2.2. Central Limit Theorem......Page 54 2.3. Cramér’s Theorem for Large Deviations......Page 55 2.4. Statistics of Extrema......Page 63 Exercises......Page 65 Notes......Page 67 Chapter 3. Markov Chains......Page 68 3.1. Discrete Time Finite Markov Chains......Page 69 3.2. Invariant Distribution......Page 71 3.3. Ergodic Theorem for Finite Markov Chains......Page 74 3.4. Poisson Processes......Page 76 3.5. ��-processes......Page 77 3.6. Embedded Chain and Irreducibility......Page 80 3.8. Time Reversal......Page 82 3.9. Hidden Markov Model......Page 84 3.10. Networks and Markov Chains......Page 90 Exercises......Page 94 Notes......Page 96 Chapter 4. Monte Carlo Methods......Page 98 4.1. Numerical Integration......Page 99 4.2. Generation of Random Variables......Page 100 4.3. Variance Reduction......Page 106 4.4. The Metropolis Algorithm......Page 110 4.5. Kinetic Monte Carlo......Page 114 4.6. Simulated Tempering......Page 115 4.7. Simulated Annealing......Page 117 Exercises......Page 119 Notes......Page 121 Chapter 5. Stochastic Processes......Page 124 5.1. Axiomatic Construction of Stochastic Process......Page 125 5.2. Filtration and Stopping Time......Page 127 5.3. Markov Processes......Page 129 5.4. Gaussian Processes......Page 132 Exercises......Page 136 Notes......Page 137 Chapter 6. Wiener Process......Page 140 6.1. The Diffusion Limit of Random Walks......Page 141 6.2. The Invariance Principle......Page 143 6.3. Wiener Process as a Gaussian Process......Page 144 6.4. Wiener Process as a Markov Process......Page 148 6.5. Properties of the Wiener Process......Page 149 6.6. Wiener Process under Constraints......Page 153 6.7. Wiener Chaos Expansion......Page 155 Exercises......Page 158 Notes......Page 160 Chapter 7. Stochastic Differential Equations......Page 162 7.1. Itô Integral......Page 163 7.2. Itô’s Formula......Page 167 7.3. Stochastic Differential Equations......Page 171 7.4. Stratonovich Integral......Page 177 7.5. Numerical Schemes and Analysis......Page 179 7.6. Multilevel Monte Carlo Method......Page 185 Exercises......Page 188 Notes......Page 190 Chapter 8. Fokker-Planck Equation......Page 192 8.1. Fokker-Planck Equation......Page 193 8.2. Boundary Condition......Page 196 8.3. The Backward Equation......Page 198 8.4. Invariant Distribution......Page 199 8.5. The Markov Semigroup......Page 201 8.6. Feynman-Kac Formula......Page 203 8.7. Boundary Value Problems......Page 204 8.8. Spectral Theory......Page 206 8.9. Asymptotic Analysis of SDEs......Page 208 8.10. Weak Convergence......Page 211 Exercises......Page 216 Notes......Page 217 Part 2 . Advanced Topics......Page 220 Chapter 9. Path Integral......Page 222 9.1. Formal Wiener Measure......Page 223 9.2. Girsanov Transformation......Page 226 9.3. Feynman-Kac Formula Revisited......Page 230 Notes......Page 231 Chapter 10. Random Fields......Page 232 10.1. Examples of Random Fields......Page 233 10.2. Gaussian Random Fields......Page 235 10.3. Gibbs Distribution and Markov Random Fields......Page 237 Notes......Page 239 Chapter 11. Introduction to Statistical Mechanics......Page 240 11.1. Thermodynamic Heuristics......Page 242 11.2. Equilibrium Statistical Mechanics......Page 247 11.3. Generalized Langevin Equation......Page 256 11.4. Linear Response Theory......Page 259 11.5. The Mori-Zwanzig Reduction......Page 261 11.6. Kac-Zwanzig Model......Page 263 Exercises......Page 265 Notes......Page 267 Chapter 12. Rare Events......Page 268 12.1. Metastability and Transition Events......Page 269 12.2. WKB Analysis......Page 271 12.3. Transition Rates......Page 272 12.4. Large Deviation Theory and Transition Paths......Page 273 12.5. Computing the Minimum Energy Paths......Page 276 12.6. Quasipotential and Energy Landscape......Page 277 Exercises......Page 282 Notes......Page 283 Chapter 13. Introduction to Chemical Reaction Kinetics......Page 284 13.1. Reaction Rate Equations......Page 285 13.2. Chemical Master Equation......Page 286 13.3. Stochastic Differential Equations......Page 288 13.5. The Large Volume Limit......Page 289 13.6. Diffusion Approximation......Page 291 13.7. The Tau-leaping Algorithm......Page 292 13.8. Stationary Distribution......Page 294 13.9. Multiscale Analysis of a Chemical Kinetic System......Page 295 Notes......Page 300 A. Laplace Asymptotics and Varadhan’s Lemma......Page 302 B. Gronwall’s Inequality......Page 304 C. Measure and Integration......Page 305 D. Martingales......Page 307 E. Strong Markov Property......Page 308 F. Semigroup of Operators......Page 309 Bibliography......Page 312 Index......Page 324 Back Cover......Page 329