Stochastic processes for physicists: Understanding noisy systems

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 9
Preface......Page 13
Thanks to . . .......Page 15
1.1 Random variables and mutually exclusive events......Page 17
1.2 Independence......Page 20
1.3 Dependent random variables......Page 21
1.4 Correlations and correlation coefficients......Page 22
1.5 Adding random variables together......Page 23
1.6 Transformations of a random variable......Page 24
1.8 The characteristic function......Page 26
1.9 Moments and cumulants......Page 28
1.10 The multivariate Gaussian......Page 29
Exercises......Page 30
2.1 Introduction......Page 32
2.2 Vector differential equations......Page 33
2.3 Writing differential equations using differentials......Page 34
2.4.1 A linear differential equation with driving......Page 36
2.5 Solving vector linear differential equations......Page 37
2.6 Diagonalizing a matrix......Page 39
Exercises......Page 40
3.1 Introduction......Page 42
3.2 Gaussian increments and the continuum limit......Page 44
3.3 Interlude: why Gaussian noise?......Page 47
3.4 Ito calculus......Page 48
3.5 Ito’s formula: changing variables in an SDE......Page 51
3.6.1 The Ornstein–Uhlenbeck process......Page 53
3.6.2 The full linear stochastic equation......Page 55
3.6.3 Ito stochastic integrals......Page 56
3.7 Deriving equations for the means and variances......Page 57
3.8.1 Stochastic equations with multiple noise sources......Page 58
3.8.2 Ito’s formula for multiple variables......Page 60
3.8.3 Multiple Ito stochastic integrals......Page 61
3.8.5 The full multivariate linear stochastic equation......Page 64
3.9 Non-anticipating functions......Page 67
Exercises......Page 68
4.1 Sample paths......Page 71
4.2 The reflection principle and the first-passage time......Page 73
4.3 The stationary auto-correlation function, g(τ)......Page 75
4.4 Conditional probability densities......Page 76
4.5 The power spectrum......Page 77
4.5.1 Signals with finite energy......Page 79
4.5.2 Signals with finite power......Page 81
4.6 White noise......Page 82
Exercises......Page 85
5.1 Physics: Brownian motion......Page 87
5.2 Finance: option pricing......Page 90
The interest rate......Page 91
Arbitrage......Page 92
5.2.2 Deriving the Black–Scholes equation......Page 94
5.2.3 Creating a portfolio that is equivalent to an option......Page 97
5.2.4 The price of a “European” option......Page 98
5.3 Modeling multiplicative noise in real systems: Stratonovich integrals......Page 101
Exercises......Page 105
6.1 Euler’s method......Page 107
6.2 Checking the accuracy of a solution......Page 108
6.3 The accuracy of a numerical method......Page 110
6.4.1 Vector equations with scalar noise......Page 111
6.4.2 Vector equations with commutative noise......Page 112
6.4.3 General vector equations......Page 113
6.5 Runge–Kutter-like methods......Page 114
6.7 Weak solutions......Page 115
6.7.1 Second-order weak methods......Page 116
Further reading......Page 117
7.1 Deriving the Fokker–Planck equation......Page 118
7.2 The probability current......Page 120
7.3 Absorbing and reflecting boundaries......Page 121
7.4 Stationary solutions for one dimension......Page 122
7.5 Physics: thermalization of a single particle......Page 123
7.6 Time-dependent solutions......Page 125
7.6.1 Green’s functions......Page 126
7.7.1 The time to exit an interval......Page 127
7.7.2 The time to exit through one end of an interval......Page 129
The mean exit time through one end......Page 131
7.8 Chemistry: reaction–diffusion equations......Page 132
7.9 Chemistry: pattern formation in reaction–diffusion systems......Page 135
Further reading......Page 140
Exercises......Page 141
8.1 The Poisson process......Page 143
8.2 Stochastic equations for jump processes......Page 146
8.3 The master equation......Page 147
8.4 Moments and the generating function......Page 149
8.5 Another simple jump process: “telegraph noise”......Page 150
8.6 Solving the master equation: a more complex example......Page 152
8.7 The general form of the master equation......Page 155
8.8 Biology: predator–prey systems......Page 156
8.9 Biology: neurons and stochastic resonance......Page 160
Further reading......Page 164
Exercises......Page 165
9.1 Introduction......Page 167
9.2 The stable Levy processes......Page 168
9.2.1 Stochastic equations with the stable processes......Page 172
9.2.2 Numerical simulation......Page 173
9.3 Characterizing all the Levy processes......Page 175
9.4 Stochastic calculus for Levy processes......Page 178
9.4.1 The linear stochastic equation with a Levy process......Page 179
Exercises......Page 180
10.1 Introduction......Page 182
10.4 The collection of events forms a sigma-algebra......Page 183
10.5 The probability measure......Page 185
The Radon–Nikodym derivative......Page 186
Probability densities......Page 187
Independent variables and σ-algebras......Page 188
Conditional expectation......Page 189
10.7 Stochastic processes: filtrations and adapted processes......Page 190
10.7.1 Martingales......Page 191
10.8 Translating the modern language......Page 192
Further reading......Page 196
Appendix A: Calculating Gaussian integrals......Page 197
References......Page 200
Index......Page 202