Stochastic calculus provides a powerful description of a specific class of stochastic processes in physics and finance. However, many econophysicists struggle to understand it. This book presents the subject simply and systematically, giving graduate students and practitioners a better understanding and enabling them to apply the methods in practice. The book develops Ito calculus and Fokker–Planck equations as parallel approaches to stochastic processes, using those methods in a unified way. The focus is on nonstationary processes, and statistical ensembles are emphasized in time series analysis. Stochastic calculus is developed using general martingales. Scaling and fat tails are presented via diffusive models. Fractional Brownian motion is thoroughly analyzed and contrasted with Ito processes. The Chapman–Kolmogorov and Fokker–Planck equations are shown in theory and by example to be more general than a Markov process. The book also presents new ideas in financial economics and a critical survey of econometrics.

Abbreviations page 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

Joseph L. McCauley is Professor of Physics at the University of Houston. During his career he has contributed to several fields, including statistical physics, superfluids, nonlinear dynamics, cosmology, econophysics, economics and finance theory.

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