Stochastic Methods A Handbook for the Natural and Social Sciences

ISBN-10: 3540707123
ISBN-13: 9783540707127
Edition: 4th 2009
Authors: Crispin Gardiner
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Description: This classic text collects, in simple language & deductive form, the many formulae & methods that can be found in the scientific literature on stochastic methods. It is written without excessive mathematical rigour, yet restricted to those methods &  More...

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Book details

Edition: 4th
Copyright year: 2009
Publisher: Springer
Publication date: 1/16/2009
Binding: Hardcover
Pages: 447
Size: 6.50" wide x 9.50" long x 1.25" tall
Weight: 1.738
Language: English

This classic text collects, in simple language & deductive form, the many formulae & methods that can be found in the scientific literature on stochastic methods. It is written without excessive mathematical rigour, yet restricted to those methods & approximations thereof, that can be systematized & controlled in a quantitative way.

A Historical Introduction
Motivation
Some Historical Examples
Brownian Motion
Langevin's Equation
The Stock Market
Statistics of Returns
Financial Derivatives
The Black-Scholes Formula
Heavy Tailed Distributions
Birth-Death Processes
Noise in Electronic Systems
Shot Noise
Autocorrelation Functions and Spectra
Fourier Analysis of Fluctuating Functions: Stationary Systems
Johnson Noise and Nyquist's Theorem
Probability Concepts
Events, and Sets of Events
Probabilities
Probability Axioms
The Meaning of P(A)
The Meaning of the Axioms
Random Variables
Joint and Conditional Probabilities: Independence
Joint Probabilities
Conditional Probabilities
Relationship Between Joint Probabilities of Different Orders
Independence
Mean Values and Probability Density
Determination of Probability Density by Means of Arbitrary Functions
Sets of Probability Zero
The Interpretation of Mean Values
Moments, Correlations, and Covariances
The Law of Large Numbers
Characteristic Function
Cumulant Generating Function: Correlation Functions and Cumulants
Example: Cumulant of Order 4: <<X1X2X3X4>>
Significance of Cumulants
Gaussian and Poissonian Probability Distributions
The Gaussian Distribution
Central Limit Theorem
The Poisson Distribution
Limits of Sequences of Random Variables
Almost Certain Limit
Mean Square Limit (Limit in the Mean)
Stochastic Limit, or Limit in Probability
Limit in Distribution
Relationship Between Limits
Markov Processes
Stochastic Processes
Kinds of Stochastic Process
Markov Process
Consistency-the Chapman-Kolmogorov Equation
Discrete State Spaces
More General Measures
Continuity in Stochastic Processes
Mathematical Definition of a Continuous Markov Process
Differential Chapman-Kolmogorov Equation
Derivation of the Differential Chapman-Kolmogorov Equation
Status of the Differential Chapman-Kolmogorov Equation
Interpretation of Conditions and Results
Jump Processes: The Master Equation
Diffusion Processes-the Fokker-Planck Equation
Deterministic Processes-Liouville's Equation
General Processes
Equations for Time Development in Initial Time-Backward Equations
Stationary and Homogeneous Markov Processes
Ergodic Properties
Homogeneous Processes
Approach to a Stationary Process
Autocorrelation Function for Markov Processes
Examples of Markov Processes
The Wiener Process
The Random Walk in One Dimension
Poisson Process
The Ornstein-Uhlenbeck Process
Random Telegraph Process
The Ito Calculus and Stochastic Differential Equations
Motivation
Stochastic Integration
Definition of the Stochastic Integral
Ito Stochastic Integral
Example W(t')dW(t')
The Stratonovich Integral
Nonanticipating Functions
Proof that dW(t)2 = dt and dW(t)2+N = 0
Properties of the Ito Stochastic Integral
Stochastic Differential Equations (SDE)
Ito Stochastic Differential Equation: Definition
Dependence on Initial Conditions and Parameters
Markov Property of the Solution of an Ito SDE
Change of Variables: Ito's Formula
Connection Between Fokker-Planck Equation and Stochastic Differential Equation
Multivariable Systems
The Stratonovich Stochastic Integral
Definition of the Stratonovich Stochastic Integral
Stratonovich Stochastic Differential Equation
Some Examples and Solutions
Coefficients without x Dependence
Multiplicative Linear White Noise Process-Geometric Brownian Motion
Complex Oscillator with Noisy Frequency
Ornstein-Uhlenbeck Process
Conversion from Cartesian to Polar Coordinates
Multivariate Ornstein-Uhlenbeck Process
The General Single Variable Linear Equation
Multivariable Linear Equations
Time-Dependent Ornstein-Uhlenbeck Process
The Fokker-Planck Equation
Probability Current and Boundary Conditions
Classification of Boundary Conditions
Boundary Conditions for the Backward Fokker-Planck Equation
Fokker-Planck Equation in One Dimension
Boundary Conditions in One Dimension
Stationary Solutions for Homogeneous Fokker-Planck Equations
Examples of Stationary Solutions
Eigenfunction Methods for Homogeneous Processes
Eigenfunctions for Reflecting Boundaries
Eigenfunctions for Absorbing Boundaries
Examples
First Passage Times for Homogeneous Processes
Two Absorbing Barriers
One Absorbing Barrier
Application-Escape Over a Potential Barrier
Probability of Exit Through a Particular End of the Interval
The Fokker-Planck Equation in Several Dimensions
Change of Variables
Stationary Solutions of Many Variable Fokker-Planck Equations
Boundary Conditions
Potential Conditions
Detailed Balance
Definition of Detailed Balance
Detailed Balance for a Markov Process
Consequences of Detailed Balance for Stationary Mean, Autocorrelation Function and Spectrum
Situations in Which Detailed Balance must be Generalised
Implementation of Detailed Balance in the Differential Chapman-Kolmogorov Equation
Examples of Detailed Balance in Fokker-Planck Equations
Kramers' Equation for Brownian Motion in a Potential
Deterministic Motion
Detailed Balance in Markovian Physical Systems
Ornstein-Uhlenbeck Process
The Onsager Relations
Significance of the Onsager Relations-Fluctuation-Dissipation Theorem
Eigenfunction Methods in Many Variables
Relationship between Forward and Backward Eigenfunctions
Even Variables Only-Negativity of Eigenvalues
A Variational Principle
Conditional Probability
Autocorrelation Matrix
Spectrum Matrix
First Exit Time from a Region (Homogeneous Processes)
Solutions of Mean Exit Time Problems
Distribution of Exit Points
Small Noise Approximations for Diffusion Processes
Comparison of Small Noise Expansions for Stochastic Differential Equations and Fokker-Planck Equations
Small Noise Expansions for Stochastic Differential Equations
Validity of the Expansion
Stationary Solutions (Homogeneous Processes)
Mean, Variance, and Time Correlation Function
Failure of Small Noise Perturbation Theories
Small Noise Expansion of the Fokker-Planck Equation
Equations for Moments and Autocorrelation Functions
Example
Asymptotic Method for Stationary Distribution
The White Noise Limit
White Noise Process as a Limit of Nonwhite Process
Formulation of the Limit
Generalisations of the Method
Brownian Motion and the Smoluchowski Equation
Systematic Formulation in Terms of Operators and Projectors
Short-Time Behaviour
Boundary Conditions
Evaluation of Higher Order Corrections
Adiabatic Elimination of Fast Variables: The General Case
Example: Elimination of Short-Lived Chemical Intermediates
Adiabatic Elimination in Haken's Model
Adiabatic Elimination of Fast Variables: A Nonlinear Case
An Example with Arbitrary Nonlinear Coupling
Beyond the White Noise Limit
Specification of the Problem
Eigenfunctions of L1
Projectors
Bloch's Perturbation Theory
Formalism for the Perturbation Theory
Application of Bloch's Perturbation Theory
Construction of the Conditional Probability
Stationary Solution Ps(x,p)
Examples
Generalisation to a system driven by several Markov Processes
Computation of Correlation Functions
Special Results for Ornstein-Uhlenbeck p(t)
Generalisation to Arbitrary Gaussian Inputs
The White Noise Limit
Relation of the White Noise Limit of <x(t)(0)> to the Impulse Response Function
Levy Processes and Financial Applications
Stochastic Description of Stock Prices
The Brownian Motion Description of Financial Markets
Financial Assets
"Long" and "Short" Positions
Perfect Liquidity
The Black-Scholes Formula
Explicit Solution for the Option Price
Analysis of the Formula
The Risk-Neutral Formulation
Change of Measure and Girsanov's Theorem
Heavy Tails and Levy Processes
Levy Processes
Infinite Divisibility
The Poisson Process
The Compound Poisson Process
Levy Processes with Infinite Intensity
The Levy-Khinchin Formula
The Paretian Processes
Shapes of the Paretian Distributions
The Events of a Paretian Process
Stable Processes
Other Levy processes
Modelling the Empirical Behaviour of Financial Markets
Stylised Statistical Facts on Asset Returns
The Paretian Process Description
Implications for Realistic Models
Equivalent Martingale Measure
Hyperbolic Models
Choice of Models
Epilogue-the Crash of 2008
Master Equations and Jump Processes
Birth-Death Master Equations-One Variable
Stationary Solutions
Example: Chemical Reaction X A
A Chemical Bistable System
Approximation of Master Equations by Fokker-Planck Equations
Jump Process Approximation of a Diffusion Process
The Kramers-Moyal Expansion
Van Kampen's System Size Expansion
Kurtz's Theorem
Critical Fluctuations
Boundary Conditions for Birth-Death Processes
Mean First Passage Times
Probability of Absorption
Comparison with Fokker-Planck Equation
Birth-Death Systems with Many Variables
Stationary Solutions when Detailed Balance Holds
Stationary Solutions Without Detailed Balance (Kirchoff's Solution)
System Size Expansion and Related Expansions
Some Examples
X + A 2X
X Y A
Prey-Predator System
Generating Function Equations
The Poisson Representation
Formulation of the Poisson Representation
Kinds of Poisson Representations
Real Poisson Representations
Complex Poisson Representations
The Positive Poisson Representation
Time Correlation Functions
Interpretation in Terms of Statistical Mechanics
Linearised Results
Trimolecular Reaction
Fokker-Planck Equation for Trimolecular Reaction
Third-Order Noise
Example of the Use of Third-Order Noise
Simulations Using the Positive Poisson representation
Analytic Treatment via the Deterministic Equation
Full Stochastic Case
Testing the Validity of Positive Poisson Simulations
Application of the Poisson Representation to Population Dynamics
The Logistic Model
Poisson Representation Stochastic Differential Equation
Environmental Noise
Extinction
Spatially Distributed Systems
Background
Functional Fokker-Planck Equations
Multivariate Master Equation Description
Continuum Form of Diffusion Master Equation
Combining Reactions and Diffusion
Poisson Representation Methods
Spatial and Temporal Correlation Structures
Reaction X Y
Reactions B + X C, A + X 2X
A Nonlinear Model with a Second-Order Phase Transition
Connection Between Local and Global Descriptions
Explicit Adiabatic Elimination of Inhomogeneous Modes
Phase-Space Master Equation
Treatment of Flow
Flow as a Birth-Death Process
Inclusion of Collisions-the Boltzmann Master Equation
Collisions and Flow Together
Bistability, Metastability, and Escape Problems
Diffusion in a Double-Well Potential (One Variable)
Behaviour for D = 0
Behaviour if D is Very Small
Exit Time
Splitting Probability
Decay from an Unstable State
Equilibration of Populations in Each Well
Kramers' Method
Example: Reversible Denaturation of Chymotrypsinogen
Bistability with Birth-Death Master Equations (One Variable)
Bistability in Multivariable Systems
Distribution of Exit Points
Asymptotic Analysis of Mean Exit Time
Kramers' Method in Several Dimensions
Example: Brownian Motion in a Double Potential
Simulation of Stochastic Differential Equations
The One Variable Taylor Expansion
Euler Methods
Higher Orders
Multiple Stochastic Integrals
The Euler Algorithm
Milstein Algorithm
The Meaning of Weak and Strong Convergence
Stability
Consistency
Implicit and Semi-implicit Algorithms
Vector Stochastic Differential Equations
Formulae and Notation
Multiple Stochastic Integrals
The Vector Euler Algorithm
The Vector Milstein Algorithm
The Strong Vector Semi-implicit Algorithm
The Weak Vector Semi-implicit Algorithm
Higher Order Algorithms
Stochastic Partial Differential Equations
Fourier Transform Methods
The Interaction Picture Method
Software Resources
References
Bibliography
Author Index
Symbol Index
Subject Index

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