Stochastic Calculus and Differential Equations for Physics and Finance

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

Chapter 1 Random variables and probability distributions
Chapter 2 Martingales, Markov, and nonstationarity
Chapter 3 Stochastic calculus
Chapter 4 Ito processes and Fokker–Plan
Chapter 6 Fractional Brownian motion
Chapter 7 Kolmogorov’s pdes and Chapman–Kolmogorov
Chapter 8 Non-Markov Ito processes
Chapter 9 Black–Scholes, martingales, and Feynman–Kac
Chapter 10 Stochastic calculus with martingales
Chapter 11 Statistical physics and finance: A brief history of each
Chapter 12 Introduction to new financial economics
Chapter 13 Statistical ensembles and time-series analysis
Chapter 14 Econometrics
Chapter 15 Semimartingales