Elements of Stochastic Methods

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PREFACE
TABLE OF CONTENTS
PART I: FOUNDATIONS
	CHAPTER 1: INTRODUCTION
		1.1 EVENTS AND THEIR PROBABILITIES
			1.1.1 Sets of Events
			1.1.2 Notations
			1.1.3 Probabilities
			1.1.4 Relating Probability to the Real World
		1.2 JOINT AND CONDITIONAL PROBABILITIES
			1.2.1 Joint Probabilities
			1.2.2 Relationship Between Joint Probabilities of Different Orders
			1.2.3 Conditional Probabilities
			1.2.4 Independence
		1.3 PROBABILITY NOTATIONS
			1.3.1 Probability Distribution Function
			1.3.2 Probability Density
			1.3.3 Random Variables
			1.3.4 Independent Random Variables
		1.4 MEAN VALUES OF RANDOM VARIABLES
			1.4.1 Definitions
			1.4.2 Some History
			1.4.3 The Law of Large Numbers
			1.4.4 Applicability of the Law of Large Numbers
			1.4.5 Heavy-Tailed Distributions
			1.4.6 Moments, Correlations, and Covariances
		1.5 THE CHARACTERISTIC FUNCTION
			1.5.1 Properties of the Characteristic Function
			1.5.2 Role and Significance of the Characteristic Function
		REFERENCES FOR CHAPTER 1
	CHAPTER 2: PROBABILITY DISTRIBUTIONS
		2.1 THE UNIFORM DISTRIBUTION
			2.1.1 Relation to Other Distributions
			2.1.2 Simulation Algorithm for an Arbitrary Probability Density
		2.2 THE GAUSSIAN DISTRIBUTION
			2.2.1 The Univariate Gaussian Distribution
			2.2.2 The Multivariate Gaussian Distribution
			2.2.3 Characteristic Function
			2.2.4 Higher Order Moments
			2.2.5 Cumulants
			2.2.6 Gaussian Approximations
		2.3 THE CENTRAL LIMIT THEOREM
			2.3.1 The Law of Large Numbers
			2.3.2 The Concept of the “Normal” Distribution
			2.3.3 Heavy-Tailed Distributions
		2.4 DISCRETE DISTRIBUTIONS
			2.4.1 The Generating Function
			2.4.2 The Binomial Distribution
			2.4.3 The Multinomial Distribution
			2.4.4 The Poisson Distribution
		2.A APPENDIX: SOME COMMONLY USED DISTRIBUTIONS
			2.A.1 The Geometric Distribution
			2.A.2 The Gamma Distribution
			2.A.3 The Inverse Gaussian Distribution
			2.A.4 The Weibull Distribution
		REFERENCES FOR CHAPTER 2
	CHAPTER 3: HEAVY-TAILED DISTRIBUTIONS
		3.1 PARETO’S LAW
			3.1.1 The Pareto Distribution Function
			3.1.2 Power-Law Distributions
			3.1.3 Pareto’s Law
		3.2 RANKED DATA AND ZIPF’S LAW
			3.2.1 Applying Zipf’s Law
			3.2.2 Comparison with Pareto’s Law
		3.3 THE LOG-NORMAL DISTRIBUTION
			3.3.1 How a Log-Normal Description Might Arise
			3.3.2 Properties of the Log-Normal Distribution
		3.4 STABLE DISTRIBUTIONS
			3.4.1 The Cauchy Distribution
			3.4.2 Definition of a Stable Distribution
			3.4.3 The Paretian Distributions
			3.4.4 The Standard Paretian Parameterization
			3.4.5 Dependence of the Paretian Distributions on the Parameters
			3.4.6 The Centered Paretian Distributions
			3.4.7 Stability, Scaling, and Width
			3.4.8 Generalized Central Limit Theorem
		3.A APPENDIX: BEHAVIOR OF PARETIAN DISTRIBUTIONS WHEN β=1
		REFERENCES FOR CHAPTER 3
PART II: STOCHASTIC PROCESSES
	CHAPTER 4: MARKOV PROCESSES
		4.1 THE MARKOV POSTULATE
			4.1.1 The Chapman–Kolmogorov Equation
			4.1.2 The Chapman–Kolmogorov Equation for Continuous Variables
			4.1.3 The Chapman–Kolmogorov Equation for Discrete Variables
		4.2 THE DIFFERENTIAL CHAPMAN–KOLMOGOROV EQUATION
			4.2.1 Time-Homogeneous Markov Processes
			4.2.2 Diffusion Processes
			4.2.3 Interpretation of the Drift Term
			4.2.4 Interpretation of the Diffusion Term
			4.2.5 Interpretation of the Jump Term
		4.3 ALGORITHM FOR SIMULATING A MARKOV PROCESS
		REFERENCES FOR CHAPTER 4
	CHAPTER 5: STATIONARY PROCESSES
		5.1 STATIONARY MARKOV PROCESSES
			5.1.1 The Stationary Differential Chapman–Kolmogorov Equation
			5.1.2 The Approach to a Stationary Solution
		5.2 MEASUREMENT IN FLUCTUATING SYSTEMS
			5.2.1 Quantities Measured in Practice
			5.2.2 The Fluctuation Correlation Function
		5.3 THE REGRESSION THEOREM
		5.4 SPECTRUM AND AUTOCORRELATION FUNCTION
			5.4.1 Conventions for the Definition of the Autocorrelation Function
			5.4.2 The Spectrum
		5.5 FOURIER ANALYSIS OF FLUCTUATING FUNCTIONS
			5.5.1 Correlation Functions of Fourier Transform Variables
		5.6 WHITE NOISE
			5.6.1 Gaussian White Noise
			5.6.2 Existence and Interpretation of the Concept of White Noise
			5.6.3 Non-Gaussian White Noise
		REFERENCES FOR CHAPTER 5
PART III: DIFFUSION PROCESSES
	CHAPTER 6: THE WIENER PROCESS AND BROWNIAN MOTION
		6.1 PHYSICAL BROWNIAN MOTION
			6.1.1 Langevin’s Equations
			6.1.2 The Langevin Source
			6.1.3 Solution of Langevin’s Equations
			6.1.4 Fluctuation-Dissipation Theorem
			6.1.5 Diffusion as a Result of Brownian Motion
			6.1.6 Interpretation as a Wiener Process
		6.2 THE WIENER PROCESS—MATHEMATICAL BROWNIAN MOTION
			6.2.1 Wiener Process for One Variable
			6.2.2 Joint Probability and Independence of Increment
			6.2.3 Sample Paths of the Wiener Process
			6.2.4 Autocorrelation Functions
			6.2.5 Multivariable Wiener Process
			6.2.6 Simulating Physical Brownian Motion
		6.3 INTERPOLATING A WIENER PROCESS—THE BROWNIAN BRIDGE
		REFERENCES FOR CHAPTER 6
	CHAPTER 7: STOCHASTIC DIFFERENTIAL EQUATIONS
		7.1 ITO STOCHASTIC DIFFERENTIAL EQUATION
			7.1.1 Calculus of Stochastic Differential Equations
			7.1.2 Change of Variables: Ito’s Formula
		7.2 THE FOKKER–PLANCK EQUATION
		7.3 THE STRATONOVICH STOCHASTIC DIFFERENTIAL EQUATION
			7.3.1 Change of Variables for the Stratonovich Stochastic Differential Equation
			7.3.2 Equivalent Stratonovich and Ito Stochastic Differential Equations
			7.3.3 Comparison of Ito and Stratonovich Formalisms
		7.4 SYSTEMS WITH MANY VARIABLES
			7.4.1 Properties of the Noise Matrix B
			7.4.2 Fokker–Planck Equations with Many Variables
			7.4.3 Brownian Motion in a Potential
		7.5 NUMERICAL SIMULATION OF STOCHASTIC DIFFERENTIAL EQUATIONS
			7.5.1 Basic Ideas
			7.5.2 Higher-Order Algorithms
			7.5.3 Resources
		REFERENCES FOR CHAPTER 7
	CHAPTER 8: THE FOKKER–PLANCK EQUATION
		8.1 FOKKER–PLANCK EQUATION IN ONE DIMENSION
			8.1.1 Simple Properties
			8.1.2 The Backward Fokker–Planck Equation
			8.1.3 Deterministic Motion
			8.1.4 The Wiener Process
		8.2 BOUNDARY CONDITIONS
			8.2.1 The Ornstein–Uhlenbeck Process
			8.2.2 Reflecting Boundary Condition—Diffusion in a Gravitational Field
			8.2.3 Reflecting Boundary Condition
			8.2.4 Absorbing Boundary Condition
			8.2.5 General Boundary Conditions
		8.3 EIGENFUNCTION METHODS FOR HOMOGENEOUS PROCESSES
			8.3.1 Eigenfunctions for Reflecting Boundaries
			8.3.2 Eigenfunctions for Absorbing Boundaries
		8.4 MANY-VARIABLE FOKKER–PLANCK EQUATIONS
			8.4.1 Boundary Conditions
			8.4.2 Stationary Solutions and Potential Conditions
			8.4.3 Eigenfunctions for the Many-Variable Fokker–Planck Equation
		REFERENCES FOR CHAPTER 8
	CHAPTER 9: ELEMENTARY DIFFUSION PROCESSES
		9.1 SINGLE-VARIABLE PROCESSES
			9.1.1 The Ornstein–Uhlenbeck Process
			9.1.2 The One-Dimensional Ornstein–Uhlenbeck Process
			9.1.3 Geometric Brownian Motion
		9.2 PROCESSES WITH MANY VARIABLES
			9.2.1 Many-Variable Ornstein–Uhlenbeck Process
			9.2.2 Solution of the Equation
			9.2.3 Stationary Solutions
			9.2.4 The Diffusive Case
			9.2.5 Example—Johnson Noise
		9.3 COMPLEX VARIABLE OSCILLATOR PROCESSES
			9.3.1 Line Broadening in a Random Frequency Oscillator
			9.3.2 The Thermalized Oscillator
			9.3.3 Equations for Phase and Amplitude
		REFERENCES FOR CHAPTER 9
	CHAPTER 10: APPROXIMATING DIFFUSION PROCESSES
		10.1 THE LIMIT OF SMALL NOISE
			10.1.1 First-Order Approximation in the General Case
			10.1.2 Stationary Solutions
		10.2 SMALL NOISE IN MULTIVARIABLE PROCESSES
			10.2.1 Example—the van der Pol Laser Equation
		10.3 ADIABATIC ELIMINATION OF FAST VARIABLES
			10.3.1 The Smoluchowski Equation
			10.3.2 General Formulation of Adiabatic Elimination of Fast Variables
		10.4 GAUSSIAN APPROXIMATIONS
		REFERENCES FOR CHAPTER 10
	CHAPTER 11: ESCAPE AND EXTINCTION
		11.1 EXIT TIMES FOR DIFFUSION PROCESSES
			11.1.1 The Distribution Function for Exit Times
			11.1.2 The Mean First Exit Time
			11.1.3 Solutions of the Equations
		11.2 APPLICATIONS
			11.2.1 Brownian Particle in a Gravitational Field
			11.2.2 Escape Over a Potential Barrier
		REFERENCES FOR CHAPTER 11
PART IV: JUMP PROCESSES
	CHAPTER 12: ELEMENTARY JUMP PROCESSES
		12.1 THE MASTER EQUATION
			12.1.1 Simulating a Jump Process
		12.2 THE POISSON PROCESS
			12.2.1 Solution Using the Generating Function
			12.2.2 The Compensated Poisson Process
			12.2.3 The Compound Poisson Process
		12.3 THE RANDOM TELEGRAPH PROCESS
			12.3.1 Example—Simulating Jumps in a Two-Level Atom
		12.4 THE CONTINUOUS TIME RANDOM WALK
			12.4.1 Solution Using the Characteristic Function
			12.4.2 Approximate Fokker–Planck Equation
		12.5 THE KRAMERS–MOYAL EXPANSION
			12.5.1 Approximate Fokker–Planck Equation
			12.5.2 Van Kampen’s System Size Expansion
			12.5.3 Using the Kramers–Moyal Approximation
		REFERENCES FOR CHAPTER 12
	CHAPTER 13: POPULATION PROCESSES
		13.1 BIRTH–DEATH MASTER EQUATIONS—ONE VARIABLE
			13.1.1 Stationary Solutions
			13.1.2 Example: Chemical Reaction X
			A
		13.2 BIRTH-DEATH SYSTEMS WITH MANY VARIABLES
			13.2.1 Combinatorial Kinetics—General Formulation
			13.2.2 Combinatorial Master Equation
			13.2.3 Stationary Solutions
			13.2.4 Stationary Solutions with Multiple Reactions
		13.3 THE SYSTEM SIZE EXPANSION FOR POPULATION PROCESSES
			13.3.1 The Chemical Fokker–Planck Equation
			13.3.2 System Size Expansion
		13.4 THE GILLESPIE ALGORITHM
			13.4.1 Improved Gillespie Algorithms
			13.4.2 Efficient Accurate Algorithms
		REFERENCES FOR CHAPTER 13
	CHAPTER 14: MODELING POPULATION PROCESSES
		14.1 INFECTIOUS DISEASES
			14.1.1 Constructing a Model of the System
		14.2 THE KERMACK–McKENDRICK EPIDEMIC MODEL
			14.2.1 Processes of Infection, Recovery and Death
			14.2.2 Deterministic Differential Equations
			14.2.3 Stochastic Modeling of the Kermack–McKendrick Model
			14.2.4 Elaborations of the Model
			14.2.5 Modeling Realistic Systems
		14.3 CELL AND SYSTEMS BIOLOGY
			14.3.1 Deterministic Equations
			14.3.2 Stochastic Representation of the Michaelis–Menten Mechanism
			14.3.3 Bursting
		REFERENCES FOR CHAPTER 14
PART V: HEAVY-TAILED PROCESSES
	CHAPTER 15: LÉVY PROCESSES
		15.1 DEFINITION OF LÉVY PROCESSES
			15.1.1 Increments of a Lévy Process
			15.1.2 Characteristic Function of a Lévy Process
			15.1.3 Classification of Lévy Processes
		15.2 LÉVY PROCESSES WITH INFINITE INTENSITY
			15.2.1 Defining and Using the Principal Value Integral
			15.2.2 Forms of the Characteristic Function
			15.2.3 The Lévy–Khinchin Formula
		15.3 THE PARETIAN PROCESSES
			15.3.1 Particular Cases of Paretian Processes
			15.3.2 Shapes of the Paretian Distributions and the “Continuous Parameterization”
			15.3.3 Scaling and Stability
			15.3.4 Processes of the Inverse Gaussian Type
		15.4 USING THE PARETIAN PROCESSES
			15.4.1 Simulating the Paretian Processes
			15.4.2 Lévy Flights
		15.5 LÉVY STOCHASTIC DIFFERENTIAL EQUATIONS
			15.5.1 The Fractional Fokker–Planck Equation
		15.6 APPLYING LÉVY PROCESSES
		REFERENCES FOR CHAPTER 15
	CHAPTER 16: FRACTIONAL BROWNIAN MOTION
		16.1 THE WORK OF H. E. HURST ON RIVER FLOWS
			16.1.1 A Simple Model of Dam Storage
			16.1.2 Data on River Flows and Related Phenomena
			16.1.3 Modeling the Observed Data
			16.1.4 Evaluating the Coefficient g for Hurst’s Data
		16.2 FRACTIONAL BROWNIAN MOTION
			16.2.1 The Scaling Property
			16.2.2 Increments of Fractional Brownian Motion
			16.2.3 The Spectrum of dBH (t )
			16.2.4 Fractional Brownian Motion Represents Three Different Kinds of Behavior
			16.2.5 Comparison with the Paretian Process
		16.3 MODELING FRACTIONAL BROWNIAN MOTION
			16.3.1 Modeling in Terms of Ornstein–Uhlenbeck Processes
			16.3.2 Mandelbrot’s Representation
			16.3.3 Time Scale Representation
		16.4 CONCLUSION
		REFERENCES FOR CHAPTER 16
AUTHOR INDEX
SUBJECT INDEX