Stochastic Methods

by Gardiner, Crispin

• ISBN13: 9783540707127
• ISBN10: 3540707123
• Edition: 4th
• Format: Hardcover
• Copyright: 2009-03-16
• Publisher: Springer Verlag

Summary

This fourth edition of the classic text "A Handbook of Stochastic Methods" has been significantly augmented, thoroughly revised, and restructured to accomodate the new material within a systematic logical framework. This new edition adheres the original aim: "to make available in simple language and deductive form, the many formulae and methods that can be found in the literature on stochastic methods."

Table of Contents

A Historical Introduction	p. 1
Motivation	p. 1
Some Historical Examples	p. 2
Brownian Motion	p. 2
Langevin's Equation	p. 6
The Stock Market	p. 8
Statistics of Returns	p. 8
Financial Derivatives	p. 9
The Black-Scholes Formula	p. 10
Heavy Tailed Distributions	p. 10
Birth-Death Processes	p. 11
Noise in Electronic Systems	p. 14
Shot Noise	p. 14
Autocorrelation Functions and Spectra	p. 18
Fourier Analysis of Fluctuating Functions: Stationary Systems	p. 19
Johnson Noise and Nyquist's Theorem	p. 20
Probability Concepts	p. 23
Events, and Sets of Events	p. 23
Probabilities	p. 24
Probability Axioms	p. 24
The Meaning of P(A)	p. 25
The Meaning of the Axioms	p. 25
Random Variables	p. 26
Joint and Conditional Probabilities: Independence	p. 27
Joint Probabilities	p. 27
Conditional Probabilities	p. 27
Relationship Between Joint Probabilities of Different Orders	p. 28
Independence	p. 28
Mean Values and Probability Density	p. 29
Determination of Probability Density by Means of Arbitrary Functions	p. 30
Sets of Probability Zero	p. 30
The Interpretation of Mean Values	p. 31
Moments, Correlations, and Covariances	p. 32
The Law of Large Numbers	p. 32
Characteristic Function	p. 33
Cumulant Generating Function: Correlation Functions and Cumulants	p. 34
Example: Cumulant of Order 4: <>	p. 36
Significance of Cumulants	p. 36
Gaussian and Poissonian Probability Distributions	p. 37
The Gaussian Distribution	p. 37
Central Limit Theorem	p. 38
The Poisson Distribution	p. 39
Limits of Sequences of Random Variables	p. 40
Almost Certain Limit	p. 40
Mean Square Limit (Limit in the Mean)	p. 41
Stochastic Limit, or Limit in Probability	p. 41
Limit in Distribution	p. 41
Relationship Between Limits	p. 41
Markov Processes	p. 42
Stochastic Processes	p. 42
Kinds of Stochastic Process	p. 42
Markov Process	p. 43
Consistency-the Chapman-Kolmogorov Equation	p. 44
Discrete State Spaces	p. 44
More General Measures	p. 45
Continuity in Stochastic Processes	p. 45
Mathematical Definition of a Continuous Markov Process	p. 46
Differential Chapman-Kolmogorov Equation	p. 47
Derivation of the Differential Chapman-Kolmogorov Equation	p. 48
Status of the Differential Chapman-Kolmogorov Equation	p. 51
Interpretation of Conditions and Results	p. 51
Jump Processes: The Master Equation	p. 51
Diffusion Processes-the Fokker-Planck Equation	p. 52
Deterministic Processes-Liouville's Equation	p. 54
General Processes	p. 55
Equations for Time Development in Initial Time-Backward Equations	p. 55
Stationary and Homogeneous Markov Processes	p. 56
Ergodic Properties	p. 57
Homogeneous Processes	p. 60
Approach to a Stationary Process	p. 60
Autocorrelation Function for Markov Processes	p. 63
Examples of Markov Processes	p. 65
The Wiener Process	p. 65
The Random Walk in One Dimension	p. 68
Poisson Process	p. 71
The Ornstein-Uhlenbeck Process	p. 72
Random Telegraph Process	p. 75
The Ito Calculus and Stochastic Differential Equations	p. 77
Motivation	p. 77
Stochastic Integration	p. 79
Definition of the Stochastic Integral	p. 79
Ito Stochastic Integral	p. 81
Example W(t')dW(t')	p. 81
The Stratonovich Integral	p. 82
Nonanticipating Functions	p. 82
Proof that dW(t)2 = dt and dW(t)2+N = 0	p. 83
Properties of the Ito Stochastic Integral	p. 85
Stochastic Differential Equations (SDE)	p. 88
Ito Stochastic Differential Equation: Definition	p. 89
Dependence on Initial Conditions and Parameters	p. 91
Markov Property of the Solution of an Ito SDE	p. 92
Change of Variables: Ito's Formula	p. 92
Connection Between Fokker-Planck Equation and Stochastic Differential Equation	p. 93
Multivariable Systems	p. 95
The Stratonovich Stochastic Integral	p. 96
Definition of the Stratonovich Stochastic Integral	p. 96
Stratonovich Stochastic Differential Equation	p. 96
Some Examples and Solutions	p. 99
Coefficients without x Dependence	p. 99
Multiplicative Linear White Noise Process-Geometric Brownian Motion	p. 100
Complex Oscillator with Noisy Frequency	p. 101
Ornstein-Uhlenbeck Process	p. 103
Conversion from Cartesian to Polar Coordinates	p. 104
Multivariate Ornstein-Uhlenbeck Process	p. 105
The General Single Variable Linear Equation	p. 108
Multivariable Linear Equations	p. 110
Time-Dependent Ornstein-Uhlenbeck Process	p. 111
The Fokker-Planck Equation	p. 113
Probability Current and Boundary Conditions	p. 114
Classification of Boundary Conditions	p. 116
Boundary Conditions for the Backward Fokker-Planck Equation	p. 116
Fokker-Planck Equation in One Dimension	p. 117
Boundary Conditions in One Dimension	p. 118
Stationary Solutions for Homogeneous Fokker-Planck Equations	p. 120
Examples of Stationary Solutions	p. 122
Eigenfunction Methods for Homogeneous Processes	p. 124
Eigenfunctions for Reflecting Boundaries	p. 124
Eigenfunctions for Absorbing Boundaries	p. 126
Examples	p. 126
First Passage Times for Homogeneous Processes	p. 130
Two Absorbing Barriers	p. 130
One Absorbing Barrier	p. 132
Application-Escape Over a Potential Barrier	p. 133
Probability of Exit Through a Particular End of the Interval	p. 135
The Fokker-Planck Equation in Several Dimensions	p. 138
Change of Variables	p. 138
Stationary Solutions of Many Variable Fokker-Planck Equations	p. 140
Boundary Conditions	p. 140
Potential Conditions	p. 141
Detailed Balance	p. 142
Definition of Detailed Balance	p. 142
Detailed Balance for a Markov Process	p. 143
Consequences of Detailed Balance for Stationary Mean, Autocorrelation Function and Spectrum	p. 144
Situations in Which Detailed Balance must be Generalised	p. 144
Implementation of Detailed Balance in the Differential Chapman-Kolmogorov Equation	p. 145
Examples of Detailed Balance in Fokker-Planck Equations	p. 149
Kramers' Equation for Brownian Motion in a Potential	p. 149
Deterministic Motion	p. 152
Detailed Balance in Markovian Physical Systems	p. 153
Ornstein-Uhlenbeck Process	p. 153
The Onsager Relations	p. 155
Significance of the Onsager Relations-Fluctuation-Dissipation Theorem	p. 156
Eigenfunction Methods in Many Variables	p. 158
Relationship between Forward and Backward Eigenfunctions	p. 159
Even Variables Only-Negativity of Eigenvalues	p. 159
A Variational Principle	p. 160
Conditional Probability	p. 161
Autocorrelation Matrix	p. 161
Spectrum Matrix	p. 162
First Exit Time from a Region (Homogeneous Processes)	p. 163
Solutions of Mean Exit Time Problems	p. 164
Distribution of Exit Points	p. 166
Small Noise Approximations for Diffusion Processes	p. 169
Comparison of Small Noise Expansions for Stochastic Differential Equations and Fokker-Planck Equations	p. 169
Small Noise Expansions for Stochastic Differential Equations	p. 171
Validity of the Expansion	p. 174
Stationary Solutions (Homogeneous Processes)	p. 175
Mean, Variance, and Time Correlation Function	p. 175
Failure of Small Noise Perturbation Theories	p. 176
Small Noise Expansion of the Fokker-Planck Equation	p. 178
Equations for Moments and Autocorrelation Functions	p. 180
Example	p. 182
Asymptotic Method for Stationary Distribution	p. 184
The White Noise Limit	p. 185
White Noise Process as a Limit of Nonwhite Process	p. 185
Formulation of the Limit	p. 186
Generalisations of the Method	p. 189
Brownian Motion and the Smoluchowski Equation	p. 192
Systematic Formulation in Terms of Operators and Projectors	p. 194
Short-Time Behaviour	p. 195
Boundary Conditions	p. 197
Evaluation of Higher Order Corrections	p. 198
Adiabatic Elimination of Fast Variables: The General Case	p. 202
Example: Elimination of Short-Lived Chemical Intermediates	p. 202
Adiabatic Elimination in Haken's Model	p. 206
Adiabatic Elimination of Fast Variables: A Nonlinear Case	p. 210
An Example with Arbitrary Nonlinear Coupling	p. 214
Beyond the White Noise Limit	p. 216
Specification of the Problem	p. 217
Eigenfunctions of L1	p. 218
Projectors	p. 218
Bloch's Perturbation Theory	p. 219
Formalism for the Perturbation Theory	p. 220
Application of Bloch's Perturbation Theory	p. 222
Construction of the Conditional Probability	p. 223
Stationary Solution Ps(x,p)	p. 226
Examples	p. 226
Generalisation to a system driven by several Markov Processes	p. 228
Computation of Correlation Functions	p. 230
Special Results for Ornstein-Uhlenbeck p(t)	p. 232
Generalisation to Arbitrary Gaussian Inputs	p. 232
The White Noise Limit	p. 233
Relation of the White Noise Limit of to the Impulse Response Function	p. 233
Levy Processes and Financial Applications	p. 235
Stochastic Description of Stock Prices	p. 235
The Brownian Motion Description of Financial Markets	p. 237
Financial Assets	p. 237
"Long" and "Short" Positions	p. 238
Perfect Liquidity	p. 238
The Black-Scholes Formula	p. 238
Explicit Solution for the Option Price	p. 240
Analysis of the Formula	p. 242
The Risk-Neutral Formulation	p. 244
Change of Measure and Girsanov's Theorem	p. 245
Heavy Tails and Levy Processes	p. 248
Levy Processes	p. 248
Infinite Divisibility	p. 249
The Poisson Process	p. 250
The Compound Poisson Process	p. 250
Levy Processes with Infinite Intensity	p. 251
The Levy-Khinchin Formula	p. 252
The Paretian Processes	p. 252
Shapes of the Paretian Distributions	p. 255
The Events of a Paretian Process	p. 255
Stable Processes	p. 257
Other Levy processes	p. 258
Modelling the Empirical Behaviour of Financial Markets	p. 258
Stylised Statistical Facts on Asset Returns	p. 258
The Paretian Process Description	p. 259
Implications for Realistic Models	p. 259
Equivalent Martingale Measure	p. 260
Hyperbolic Models	p. 262
Choice of Models	p. 262
Epilogue-the Crash of 2008	p. 263
Master Equations and Jump Processes	p. 264
Birth-Death Master Equations-One Variable	p. 264
Stationary Solutions	p. 265
Example: Chemical Reaction X A	p. 267
A Chemical Bistable System	p. 269
Approximation of Master Equations by Fokker-Planck Equations	p. 273
Jump Process Approximation of a Diffusion Process	p. 273
The Kramers-Moyal Expansion	p. 275
Van Kampen's System Size Expansion	p. 276
Kurtz's Theorem	p. 280
Critical Fluctuations	p. 281
Boundary Conditions for Birth-Death Processes	p. 283
Mean First Passage Times	p. 284
Probability of Absorption	p. 286
Comparison with Fokker-Planck Equation	p. 286
Birth-Death Systems with Many Variables	p. 287
Stationary Solutions when Detailed Balance Holds	p. 288
Stationary Solutions Without Detailed Balance (Kirchoff's Solution)	p. 290
System Size Expansion and Related Expansions	p. 291
Some Examples	p. 291
X + A 2X	p. 291
X Y A	p. 292
Prey-Predator System	p. 292
Generating Function Equations	p. 297
The Poisson Representation	p. 301
Formulation of the Poisson Representation	p. 301
Kinds of Poisson Representations	p. 305
Real Poisson Representations	p. 305
Complex Poisson Representations	p. 306
The Positive Poisson Representation	p. 309
Time Correlation Functions	p. 313
Interpretation in Terms of Statistical Mechanics	p. 314
Linearised Results	p. 318
Trimolecular Reaction	p. 318
Fokker-Planck Equation for Trimolecular Reaction	p. 319
Third-Order Noise	p. 323
Example of the Use of Third-Order Noise	p. 324
Simulations Using the Positive Poisson representation	p. 326
Analytic Treatment via the Deterministic Equation	p. 326
Full Stochastic Case	p. 329
Testing the Validity of Positive Poisson Simulations	p. 331
Application of the Poisson Representation to Population Dynamics	p. 332
The Logistic Model	p. 332
Poisson Representation Stochastic Differential Equation	p. 333
Environmental Noise	p. 333
Extinction	p. 334
Spatially Distributed Systems	p. 336
Background	p. 336
Functional Fokker-Planck Equations	p. 338
Multivariate Master Equation Description	p. 339
Continuum Form of Diffusion Master Equation	p. 341
Combining Reactions and Diffusion	p. 345
Poisson Representation Methods	p. 346
Spatial and Temporal Correlation Structures	p. 347
Reaction X Y	p. 347
Reactions B + X C, A + X 2X	p. 350
A Nonlinear Model with a Second-Order Phase Transition	p. 355
Connection Between Local and Global Descriptions	p. 359
Explicit Adiabatic Elimination of Inhomogeneous Modes	p. 360
Phase-Space Master Equation	p. 362
Treatment of Flow	p. 362
Flow as a Birth-Death Process	p. 363
Inclusion of Collisions-the Boltzmann Master Equation	p. 366
Collisions and Flow Together	p. 369
Bistability, Metastability, and Escape Problems	p. 372
Diffusion in a Double-Well Potential (One Variable)	p. 372
Behaviour for D = 0	p. 373
Behaviour if D is Very Small	p. 373
Exit Time	p. 375
Splitting Probability	p. 375
Decay from an Unstable State	p. 377
Equilibration of Populations in Each Well	p. 378
Kramers' Method	p. 378
Example: Reversible Denaturation of Chymotrypsinogen	p. 382
Bistability with Birth-Death Master Equations (One Variable)	p. 384
Bistability in Multivariable Systems	p. 386
Distribution of Exit Points	p. 387
Asymptotic Analysis of Mean Exit Time	p. 391
Kramers' Method in Several Dimensions	p. 392
Example: Brownian Motion in a Double Potential	p. 394
Simulation of Stochastic Differential Equations	p. 401
The One Variable Taylor Expansion	p. 402
Euler Methods	p. 402
Higher Orders	p. 402
Multiple Stochastic Integrals	p. 403
The Euler Algorithm	p. 404
Milstein Algorithm	p. 406
The Meaning of Weak and Strong Convergence	p. 407
Stability	p. 407
Consistency	p. 410
Implicit and Semi-implicit Algorithms	p. 410
Vector Stochastic Differential Equations	p. 411
Formulae and Notation	p. 412
Multiple Stochastic Integrals	p. 412
The Vector Euler Algorithm	p. 414
The Vector Milstein Algorithm	p. 414
The Strong Vector Semi-implicit Algorithm	p. 415
The Weak Vector Semi-implicit Algorithm	p. 415
Higher Order Algorithms	p. 416
Stochastic Partial Differential Equations	p. 417
Fourier Transform Methods	p. 418
The Interaction Picture Method	p. 419
Software Resources	p. 420
References	p. 421
Bibliography	p. 429
Author Index	p. 434
Symbol Index	p. 435
Subject Index	p. 439
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