Foundations of Quantitative Finance, Book VII: Brownian Motion and Other Stochastic Processes

This is the seventh book in a set of ten published under the collective title of Foundations of Quantitative Finance. The targeted readers are students, researchers, and practitioners of quantitative finance who find that many sources for financial applications are written at a level assuming significant mathematical expertise.

The goal for this series is to provide a complete and detailed development of the many foundational mathematical theories and results one finds referenced in popular resources in finance and quantitative finance. The included topics have been curated from vast mathematics and finance literature for the express purpose of supporting applications in quantitative finance. The hope is this series will advance the reader’s career.

The series is logically sequential. Books I, III, and V develop foundational mathematical results needed for the probability theory and finance applications of Books II, IV, and VI, respectively. Books VII, VIII, and IX then develop results in the theory of stochastic processes, and Book X develops applications of these stochastic and other models to finance. All ten volumes are extensively self-referenced.

Book VII introduces and develops properties of Brownian motion, arguably the most famous of stochastic processes, as well as two other classes of stochastic processes with properties enjoyed by Brownian motion, namely, Markov processes and martingales. Brownian motion is the central idea needed for most continuous time models in finance, and indeed most applications of stochastic processes. For some with more experience, this book will be the place to start in the series, while others will find Books I-VI provide a needed foundation.

Foundations of Quantitative Finance, by Robert Reitano:

Book I. Measure Spaces and Measurable Functions

Book II. Probability Spaces and Random Variables

Book III. The Integrals of Riemann, Lebesgue and (Riemann-)Stieltjes

Book IV. Distribution Functions and Expectations

Book V. General Measure and Integration Theory

Book VI. Densities, Transformed Distributions, and Limit Theorems

Book VII. Brownian Motion and Other Stochastic Processes

Book VIII. Itô Integration and Stochastic Calculus 1

Book IX. Stochastic Calculus 2 and Stochastic Differential Equations

Book X. Classical Models and Applications in Finance