by Thomas Mikosch (University of Groningen)

Modelling with the Itô integral or stochastic differential equations has become increasingly important in various applied fields, including physics, biology, chemistry and finance. However, stochastic calculus is based on a deep mathematical theory.

This book is suitable for the reader without a deep mathematical background. It gives an elementary introduction to that area of probability theory, without burdening the reader with a great deal of measure theory. Applications are taken from stochastic finance. In particular, the Black-Scholes option pricing formula is derived. The book can serve as a text for a course on stochastic calculus for non-mathematicians or as elementary reading material for anyone who wants to learn about Itô calculus and/or stochastic finance.

• 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