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Stochastic Calculus and Differential Equations for Physics and Finance

Book Description

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.

Table of Contents

  1. Coverpage
  2. Stochastic Calculus and Differential Equations for Physics and Finance
  3. Title page
  4. Copyright page
  5. Dedication
  6. Contents
  7. Abbreviations
  8. Introduction
  9. 1 Random variables and probability distributions
    1. 1.1 Particle descriptions of partial differential equations
    2. 1.2 Random variables and stochastic processes
    3. 1.3 The n-point probability distributions
    4. 1.4 Simple averages and scaling
    5. 1.5 Pair correlations and 2-point densities
    6. 1.6 Conditional probability densities
    7. 1.7 Statistical ensembles and time series
    8. 1.8 When are pair correlations enough to identify a stochastic process?
    9. Exercises
  10. 2 Martingales, Markov, and nonstationarity
    1. 2.1 Statistically independent increments
    2. 2.2 Stationary increments
    3. 2.3 Martingales
    4. 2.4 Nonstationary increment processes
    5. 2.5 Markov processes
    6. 2.6 Drift plus noise
    7. 2.7 Gaussian processes
    8. 2.8 Stationary vs. nonstationary processes
    9. Exercises
  11. 3 Stochastic calculus
    1. 3.1 The Wiener process
    2. 3.2 Ito's theorem
    3. 3.3 Ito's lemma
    4. 3.4 Martingales for greenhorns
    5. 3.5 First-passage times
    6. Exercises
  12. 4 Ito processes and Fokker–Planck equations
    1. 4.1 Stochastic differential equations
    2. 4.2 Ito's lemma
    3. 4.3 The Fokker–Planck pde
    4. 4.4 The Chapman–Kolmogorov equation
    5. 4.5 Calculating averages
    6. 4.6 Statistical equilibrium
    7. 4.7 An ergodic stationary process
    8. 4.8 Early models in statistical physics and finance
    9. 4.9 Nonstationary increments revisited
    10. Exercises
  13. 5 Selfsimilar Ito processes
    1. 5.1 Selfsimilar stochastic processes
    2. 5.2 Scaling in diffusion
    3. 5.3 Superficially nonlinear diffusion
    4. 5.4 Is there an approach to scaling?
    5. 5.5 Multiaffine scaling
    6. Exercises
  14. 6 Fractional Brownian motion
    1. 6.1 Introduction
    2. 6.2 Fractional Brownian motion
    3. 6.3 The distribution of fractional Brownian motion
    4. 6.4 Infinite memory processes
    5. 6.5 The minimal description of dynamics
    6. 6.6 Pair correlations cannot scale
    7. 6.7 Semimartingales
    8. Exercises
  15. 7 Kolmogorov's pdes and Chapman–Kolmogorov
    1. 7.1 The meaning of Kolmogorov's first pde
    2. 7.2 An example of backward-time diffusion
    3. 7.3 Deriving the Chapman–Kolmogorov equation for an Ito process
    4. Exercise
  16. 8 Non-Markov Ito processes
    1. 8.1 Finite memory Ito processes?
    2. 8.2 A Gaussian Ito process with 1-state memory
    3. 8.3 McKean's examples
    4. 8.4 The Chapman–Kolmogorov equation
    5. 8.5 Interacting system with a phase transition
    6. 8.6 The meaning of the Chapman–Kolmogorov equation
    7. Exercise
  17. 9 Black–Scholes, martingales, and Feynman–Kac
    1. 9.1 Local approximation to sdes
    2. 9.2 Transition densities via functional integrals
    3. 9.3 Black–Scholes-type pdes
    4. Exercise
  18. 10 Stochastic calculus with martingales
    1. 10.1 Introduction
    2. 10.2 Integration by parts
    3. 10.3 An exponential martingale
    4. 10.4 Girsanov's theorem
    5. 10.5 An application of Girsanov's theorem
    6. 10.6 Topological inequivalence of martingales with Wiener processes
    7. 10.7 Solving diffusive pdes by running an Ito process
    8. 10.8 First-passage times
    9. 10.9 Martingales generally seen
    10. Exercises
  19. 11 Statistical physics and finance: A brief history of each
    1. 11.1 Statistical physics
    2. 11.2 Finance theory
    3. Exercise
  20. 12 Introduction to new financial economics
    1. 12.1 Excess demand dynamics
    2. 12.2 Adam Smith's unreliable hand
    3. 12.3 Efficient markets and martingales
    4. 12.4 Equilibrium markets are inefficient
    5. 12.5 Hypothetical FX stability under a gold standard
    6. 12.6 Value
    7. 12.7 Liquidity, reversible trading, and fat tails vs. crashes
    8. 12.8 Spurious stylized facts
    9. 12.9 An sde for increments
    10. Exercises
  21. 13 Statistical ensembles and time-series analysis
    1. 13.1 Detrending economic variables
    2. 13.2 Ensemble averages and time series
    3. 13.3 Time-series analysis
    4. 13.4 Deducing dynamics from time series
    5. 13.5 Volatility measures
    6. Exercises
  22. 14 Econometrics
    1. 14.1 Introduction
    2. 14.2 Socially constructed statistical equilibrium
    3. 14.3 Rational expectations
    4. 14.4 Monetary policy models
    5. 14.5 The monetarist argument against government intervention
    6. 14.6 Rational expectations in a real, nonstationary market
    7. 14.7 Volatility, ARCH, and GARCH
    8. Exercises
  23. 15 Semimartingales
    1. 15.1 Introduction
    2. 15.2 Filtrations
    3. 15.3 Adapted processes
    4. 15.4 Martingales
    5. 15.5 Semimartingales
    6. Exercise
  24. References
  25. Index