物理学家用的随机过程【2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载】

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1 A review of probability theory1

1.1 Random variables and mutually exclusive events1

1.2 Independence4

1.3 Dependent random variables5

1.4 Correlations and correlation coefficients6

1.5 Adding independent random variables together7

1.6 Transformations ofa random variable8

1.7 The distribution function10

1.8 The characteristic function10

1.9 Moments and cumulants12

1.10 The multivariate Gaussian13

2 Differential equations16

2.1 Introduction16

2.2 Vector differential equations17

2.3 Writing differential equations using differentials18

2.4 Two methods for solving differential equations18

2.4.1 A linear differential equation with driving20

2.5 Solving vector linear differential equations21

2.6 Diagonalizing a matrix23

3 Stochastic equations with Gaussian noise26

3.1 Introduction26

3.2 Gaussian increments and the continuum limit28

3.3 Interlude:why Gaussian noise？31

3.4 Ito calculus32

3.5 Ito’s formula:changing variables in an SDE35

3.6 Solving some stochastic equations37

3.6.1 The Ornstein-Uhlenbeck process37

3.6.2 The full linear stochastic equation39

3.6.3 Ito stochastic integrals40

3.7 Deriving equations for the means and variances41

3.8 Multiple variables and multiple noise sources42

3.8.1 Stochastic equations with multiple noise sources42

3.8.2 Ito’s formula for multiple variables44

3.8.3 Multiple Ito stochastic integrals45

3.8.4 The multivariate linear equation with additive noise48

3.8.5 The full multivariate linear stochastic equation48

3.9 Non-anticipating functions51

4 Further properties of stochastic processes55

4.1 Samplepaths55

4.2 The reflection principle and the first-passage time57

4.3 The stationary auto-correlation function,g（τ）59

4.4 Conditional probability densities60

4.5 The power spectrum61

4.5.1 Signals with finite energy63

4.5.2 Signals with finite power65

4.6 White noise66

5 Some applications of Gaussian noise71

5.1 Physics:Brownian motion71

5.2 Finance:option pricing74

5.2.1 Some preliminary concepts75

5.2.2 Deriving the Black-Scholes equation78

5.2.3 Creating a portfolio that is equivalent to an option81

5.2.4 The price of a“European”option82

5.3 Modeling multiplicative noise in real systems:Stratonovich integrals85

6 Numerical methods for Gaussian noise91

6.1 Euler’s method91

6.1.1 Generating Gaussian random variables92

6.2 Checking the accuracy of a solution92

6.3 The accuracy of a numerical method94

6.4 Milstein’s method95

6.4.1 Vector equations with scalar noise95

6.4.2 Vector equations with commutative noise96

6.4.3 General vector equations97

6.5 Runge-Kutta-like methods98

6.6 Implicit methods99

6.7 Weak solutions99

6.7.1 Second-order weak methods100

7 Fokker-Planck equations and reaction-diffusion systems102

7.1 Deriving the Fokker-Planck equation102

7.2 The probability current104

7.3 Absorbing and reflecting boundaries105

7.4 Stationary solutions for one dimension106

7.5 Physics:thermalization of a single particle107

7.6 Time-dependent solutions109

7.6.1 Green’s functions110

7.7 Calculating first-passage times111

7.7.1 The time to exit an interval111

7.7.2 The time to exit through one end of an interval113

7.8 Chemistry:reaction-diffusion equations116

7.9 Chemistry:pattern formation in reaction-diffusion systems119

8 Jumpprocesses127

8.1 The Poisson process127

8.2 Stochastic equations for jump processes130

8.3 The master equation131

8.4 Moments and the generating function133

8.5 Another simple jump process:“telegraph noise”134

8.6 Solving the master equation:a more complex example136

8.7 The general form of the master equation139

8.8 Biology:predator-prey systems140

8.9 Biology:neurons and stochastic resonance144

9 Levy processes151

9.1 Introduction151

9.2 The stable Levy processes152

9.2.1 Stochastic equations with the stable processes156

9.2.2 Numerical simulation157

9.3 Characterizing all the Levy processes159

9.4 Stochastic calculus for Levy processes162

9.4.1 The linear stochastic equation with a Levy process163

10 Modern probability theory166

10.1 Introduction166

10.2 The set of all samples167

10.3 The collection of all events167

10.4 The collection of events forms a sigma-algebra167

10.5 The probability measure169

10.6 Collecting the concepts:random variables171

10.7 Stochastic processes:filtrations and adapted processes174

10.7.1 Martingales175

10.8 Translating the modern language176

Appendix A Calculating Gaussian integrals181

References184

Index186