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