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Preface Acknowledgments 1 A review of probability theory 1.1 Random variables and mutually exclusive events 1.2 Independence 1.3 Dependent random variables 1.4 Correlations and correlation coefficients 1.5 Adding independent random variables together 1.6 Transformations of a random variable 1.7 The distribution function 1.8 The characteristic function 1.9 Moments and cumulants 1.10 The multivariate Gaussian 2 Differential equations 2.1 Introduction 2.2 Vector differential equations 2.3 Writing differential equations using differentials 2.4 Two methods for solving differential equations 2.4.1 A linear differential equation with driving 2.5 Solving vector linear differential equations 2.6 Diagonalizing a matrix 3 Stochastic equations with Gaussian noise 3.1 Introduction 3.2 Gaussian increments and the continuum limit 3.3 Interlude: why Gaussian noise? 3.4 Ito calculus 3.5 Ito's formula: changing variables in an SDE 3.6 Solving some stochastic equations 3.6.1 The Ornstein-Uhlenbeck process 3.6.2 The full linear stochastic equation 3.6.3 Ito stochastic integrals 3.7 Deriving equations for the means and variances 3.8 Multiple variables and multiple noise sources 3.8.1 Stochastic equations with multiple noise sources 3.8.2 Ito's formula for multiple variables 3.8.3 Multiple Ito stochastic integrals 3.8.4 The multivariate linear equation with additive noise 3.8.5 The full multivariate linear stochastic equation 3.9 Non-anticipating functions 4 Further properties of stochastic processes 4.1 Sample paths 4.2 The reflection principle and the first-passage time 4.3 The stationary auto-correlation function, g 4.4 Conditional probability densities 4.5 The power spectrum 4.5.1 Signals with finite energy 4.5.2 Signals with finite power 4.6 White noise 5 Some applications of Gaussian noise 5.1 Physics: Brownian motion 5.2 Finance: option pricing 5.2.1 Some preliminary concepts 5.2.2 Deriving the Black-Scholes equation 5.2.3 Creating a portfolio that is equivalent to an option 5.2.4 The price of a "European" option 5.3 Modeling multiplicative noise in real systems: Stratonovich integrals 6 Numerical methods for Gaussian noise 6.1 Euler's method 6.1.1 Generating Gaussian random variables 6.2 Checking the accuracy of a solution 6.3 The accuracy of a numerical method 6.4 Milstein's method 6.4.1 Vector equations with scalar noise 6.4.2 Vector equations with commutative noise 6.4.3 General vector equations 6.5 Runge-Kutta-like methods 6.6 Implicit methods 6.7 Weak solutions 6.7.1 Second-order weak methods 7 Fokker-Planck equations and reaction--diffusion systems 7.1 Deriving the Fokker-Planck equation 7.2 The probability current 7.3 Absorbing and reflecting boundaries 7.4 Stationary solutions for one dimension 7.5 Physics: thermalization of a single particle 7.6 Time-dependent solutions 7.6.1 Green's functions 7.7 Calculating first-passage times 7.7.1 The time to exit an interval 7.7.2 The time to exit through one end of an interval 7.8 Chemistry: reaction-diffusion equations 7.9 Chemistry: pattern formation in reaction-diffusion systems 8 Jump processes 8.1 The Poisson process 8.2 Stochastic equations for jump processes 8.3 The master equation 8.4 Moments and the generating function 8.5 Another simple jump process: "telegraph noise" 8.6 Solving the master equation: a more complex example 8.7 The general form of the master equation 8.8 Biology: predator-prey systems 8.9 Biology: neurons and stochastic resonance 9 Levy processes 9.1 Introduction 9.2 The stable Levy processes 9.2.1 Stochastic equations with the stable processes 9.2.2 Numerical simulation 9.3 Characterizing all the Levy processes 9.4 Stochastic calculus for Levy processes 9.4.1 The linear stochastic equation with a Levy process 10 Modern probability theory 10.1 Introduction 10.2 The set of all samples 10.3 The collection of all events 10.4 The collection of events forms a sigma-algebra 10.5 The probability measure 10.6 Collecting the concepts: random variables 10.7 Stochastic processes: filtrations and adapted processes 10.7.1 Martingales 10.8 Translating the modem language Appendix A Calculating Gaussian integrals References Index