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Elementary Stochastic Calculus: With Finance in View

Thomas Mikosch
World Scientific
Publication Date: 
Number of Pages: 
Advanced Series on Statistical Science and Applied Probability 6
[Reviewed by
Ita Cirovic Donev
, on

When reading the title you may wonder how stochastic calculus can be elementary. Well, it really can't, it just depends on how you look at it. Elementary Stochastic Calculus is meant to provide an intuitive introduction to the subject. This is not a deep measure theoretic approach to explaining stochastic calculus. The author takes the other road — intuitive explanation with as few mathematical technicalities as possible. In only 200 pages he does indeed achieve this. Don't get me wrong, the book is technical to the extent of the actual theory of stochastic calculus. However, it does not even come close to some other texts, such as Introduction to Stochastic Calculus with Applications, by Klebaner, which is very technical.

The book assumes familiarity with calculus and elementary probability theory. The first part (almost half of the book) provides the reader with some preliminaries from probability theory and stochastic processes. The rest of the book deals with the stochastic integrals, SDEs and finally some applications of stochastic calculus in finance. Most of the important concepts are boxed, which provides a nice reference for later use.

The discussion is elegant and intuitive. There is no formal presentation of the concepts in a theorem-proof style. Technical concepts are presented without proof. There are plenty of examples scattered throughout the text. There are no official problems at the end of the chapter, but rather the author asks questions within the discussion.

What are some drawbacks of the book? Well, certainly the lack of exercises is one and probably the most serious one. As the book is intended as an introduction for beginner in stochastic calculus, there should most definitely be a list of exercises,  possibly even with solutions. Another problem, which is less serious, is the actual application to finance. There is one, very small, chapter devoted to applications. The author explains the basic aspects of option theory and moves on to the famous Black-Scholes equation. Reader more interested in applications in finance should look elsewhere.
An advanced reader with a strong mathematical background in mathematical and functional analysis and measure theory will find this book inadequate. However, this is an introductory text and as such it should be read. It is a very good prelude to books such as Klebaner's Introduction to Stochastic Calculus with Applications by Klebaner or  other more advanced texts on stochastic calculus.

Ita Cirovic Donev is a PhD candidate at the University of Zagreb. She hold a Masters degree in statistics from Rice University. Her main research areas are in mathematical finance; more precisely, statistical mehods of credit and market risk. Apart from the academic work she does consulting work for financial institutions.

  • Preliminaries:
  • Basic Concepts from Probability Theory
  • Stochastic Processes
  • Brownian Motion
  • Conditional Expectation
  • Martingales
  • The Stochastic Integral:
  • The Riemann and Riemann–Stieltjes Integrals
  • The Itô Integral
  • The Itô Lemma
  • The Stratonovich and Other Integrals
  • Stochastic Differential Equations:
  • Deterministic Differential Equations
  • Itô Stochastic Differential Equations
  • The General Linear Differential Equation
  • Numerical Solution
  • Applications of Stochastic Calculus in Finance:
  • The Black–Scholes Option-Pricing Formula
  • A Useful Technique: Change of Measure
  • Appendices:
  • Modes of Convergence
  • Inequalities
  • Non-Differentiability and Unbounded Variation of Brownian Sample Paths
  • Proof of the Existence of the General Itô Stochastic Integral
  • The Radon–Nikodym Theorem
  • Proof of the Existence and Uniqueness of the Conditional Expectation