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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 | |