Abbreviations .................................................. xi
Introduction .................................................... 1
1 Random variables and probability distributions ............... 5
1.1 Particle descriptions of partial differential
equations ............................................... 5
1.2 Random variables and stochastic processes ............... 7
1.3 The n-point probability distributions ................... 9
1.4 Simple averages and scaling ............................ 10
1.5 Pair correlations and 2-point densities ................ 11
1.6 Conditional probability densities ...................... 12
1.7 Statistical ensembles and time series .................. 13
1.8 When are pair correlations enough to identify a
stochastic process? .................................... 16
Exercises ................................................... 17
2 Martingales, Markov, and nonstationarity .................... 18
2.1 Statistically independent increments ................... 18
2.2 Stationary increments .................................. 19
2.3 Martingales ............................................ 20
2.4 Nonstationary increment processes ...................... 21
2.5 Markov processes ....................................... 22
2.6 Drift plus noise ....................................... 22
2.7 Gaussian processes ..................................... 23
2.8 Stationary vs. nonstationary processes ................. 24
Exercises ................................................... 26
3 Stochastic calculus ......................................... 28
3.1 The Wiener process ..................................... 28
3.2 Ito's theorem .......................................... 29
3.3 Ito's lemma ............................................ 30
3.4 Martingales for greenhorns ............................. 31
3.5 First-passage times .................................... 33
Exercises ................................................... 35
4 Ito processes and Fokker-Planck equations ................... 37
4.1 Stochastic differential equations ...................... 37
4.2 Ito's lemma ............................................ 39
4.3 The Fokker-Planck pde .................................. 39
4.4 The Chapman-Kolmogorov equation ........................ 41
4.5 Calculating averages ................................... 42
4.6 Statistical equilibrium ................................ 43
4.7 An ergodic stationary process .......................... 45
4.8 Early models in statistical physics and finance ........ 45
4.9 Nonstationary increments revisited ..................... 48
Exercises ................................................... 48
5 Selfsimilar Ito processes ................................... 50
5.1 Selfsimilar stochastic processes ....................... 50
5.2 Scaling in diffusion ................................... 51
5.3 Superficially nonlinear diffusion ...................... 53
5.4 Is there an approach to scaling? ....................... 54
5.5 Multiaffine scaling .................................... 55
Exercises ................................................... 56
6 Fractional Brownian motion .................................. 57
6.1 Introduction ........................................... 57
6.2 Fractional Brownian motion ............................. 57
6.3 The distribution of fractional Brownian motion ......... 60
6.4 Infinite memory processes .............................. 61
6.5 The minimal description of dynamics .................... 62
6.6 Pair correlations cannot scale ......................... 63
6.7 Semimartingales ........................................ 64
Exercises ................................................... 65
7 Kolmogorov's pdes and Chapman-Kolmogorov .................... 66
7.1 The meaning of Kolmogorov's first pde .................. 66
7.2 An example of backward-time diffusion .................. 68
7.3 Deriving the Chapman-Kolmogorov equation for
an Ito process ......................................... 68
Exercise .................................................... 70
8 Non-Markov Ito processes .................................... 71
8.1 Finite memory Ito processes? ........................... 71
8.2 A Gaussian Ito process with 1-state memory ............. 72
8.3 McKean's examples ...................................... 74
8.4 The Chapman-Kolmogorov equation ........................ 78
8.5 Interacting system with a phase transition ............. 79
8.6 The meaning of the Chapman-Kolmogorov equation ......... 81
Exercise .................................................... 82
9 Black-Scholes, martingales, and Feynman-Kac ................. 83
9.1 Local approximation to sdes ............................ 83
9.2 Transition densities via functional integrals .......... 83
9.3 Black-Scholes-type pdes ................................ 84
Exercise .................................................... 85
10 Stochastic calculus with martingales ........................ 86
10.1 Introduction ........................................... 86
10.2 Integration by parts ................................... 87
10.3 An exponential martingale .............................. 88
10.4 Girsanov's theorem ..................................... 89
10.5 An application of Girsanov's theorem ................... 91
10.6 Topological inequivalence of martingales with Wiener
processes .............................................. 93
10.7 Solving diffusive pdes by running an Ito process ....... 96
10.8 First-passage times .................................... 97
10.9 Martingales generally seen ............................ 102
Exercises .................................................. 105
11 Statistical physics and finance: A brief history of each ... 106
11.1 Statistical physics ................................... 106
11.2 Finance theory ........................................ 110
Exercise ................................................... 115
12 Introduction to new financial economics .................... 117
12.1 Excess demand dynamics ................................ 117
12.2 Adam Smith's unreliable hand .......................... 118
12.3 Efficient markets and martingales ..................... 120
12.4 Equilibrium markets are inefficient ................... 123
12.5 Hypothetical FX stability under a gold standard ....... 126
12.6 Value ................................................. 131
12.7 Liquidity, reversible trading, and fat tails vs.
crashes ............................................... 132
12.8 Spurious stylized facts ............................... 143
12.9 An sde for increments ................................. 146
Exercises .................................................. 147
13 Statistical ensembles and time-series analysis ............. 148
13.1 Detrending economic variables ......................... 148
13.2 Ensemble averages and time series ..................... 149
13.3 Time-series analysis .................................. 152
13.4 Deducing dynamics from time series .................... 162
13.5 Volatility measures ................................... 167
Exercises .................................................. 168
14 Econometrics ............................................... 169
14.1 Introduction .......................................... 169
14.2 Socially constructed statistical equilibrium .......... 172
14.3 Rational expectations ................................. 175
14.4 Monetary policy models ................................ 177
14.5 The monetarist argument against government
intervention .......................................... 179
14.6 Rational expectations in a real, nonstationary
market ................................................ 180
14.7 Volatility, ARCH, and GARCH ........................... 192
Exercises .................................................. 195
15 Semimartingales ............................................ 196
15.1 Introduction .......................................... 196
15.2 Filtrations ........................................... 197
15.3 Adapted processes ..................................... 197
15.4 Martingales ........................................... 198
15.5 Semimartingales ....................................... 198
Exercise ................................................... 199
References ................................................. 200
Index ......................................................... 204
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