Ten years ago I would not have dared to write a book like this: a non-rigorous treatment of a mathematical theory. I admit that I would have been ashamed,and I am afraid that most of my colleagues in mathematics still think like this.However, my experience with students and practitioners convinced me that there is a strong demand for popular mathematics.

Reader Guidelines

1 Preliminaries

 1.1 Basic Concepts fl'om Probability Theory

1.1.1 Random Variables

1.1.2 Random Vectors

1.1.3 Independence and Dependence

 1.2 Stochastic Processes

 1.3 Brownian Motion

1.3.1 Defining Properties

1.3.2 Processes Derived from Brownian Motion

1.3.3 Simulation of Brownian Sample Paths

 1.4 Conditional Expectation

1.4.1 Conditional Expectation under Discrete Condition

1.4.2 About a-Fields

1.4.3 The General Conditional Expectation

1.4.4 Rules for the Calculation of Conditional Expectations

1.4.5 The Projection Property of Conditional Expectations

 1.5 Martingales

1.5.1 Defining Properties

1.5.2 Examples

1.5.3 The Interpretation of a Martingale as a Fair Game

2 The Stochastic Integral

 2.1 The Riemann and Riemann-Stieltjes Integrals

2.1.1 The Ordinary Riemann Integral

2.1.2 The Riemann-Stieltjes Integral

 2.2 The Ito Integral

2.2.1 A Motivating Example

2.2.2 The Ito Stochastic Integral for Simple Processes

2.2.3 The General Ito Stochastic Integral

 2.3 The Ito Lemma

2.3.1 The Classical Chain Rule of Differentiation

2.3.2 A Simple Version of the Ito Lemma

2.3.3 Extended Versions of the Ito Lemma

 2.4 The Stratonovich and Other Integrals

3 Stochastic Differential Equations

 3.1 Deterministic Differential Equations

 3.2 Ito Stochastic Differential Equations

3.2.1 What is a Stochastic Differential Equation?

3.2.2 Solving Ito Stochastic Differential Equations by the ItoLemma

3.2.3 Solving Ito Differential Equations via Stratonovich Calculus

 3.3 The General Linear Differential Equation

3.3.1 Linear Equations with Additive Noise

3.3.2 Homogeneous Equations with Multiplicative Noise

3.3.3 The General Case

3.3.4 The Expectation and Variance Functions of the Solution

 3.4 Numerical Solution

3.4.1 The Euler Approximation

3.4.2 The Milstein Approximation

4 Applications of Stochastic Calculus in Finance

 4.1 The Black-Scholes Option Pricing Formula

4.1.1 A Short Excursion into Finance

4.1.2 What is an Option?

4.1.3 A Mathematical Formulation of the Option Pricing Problem

4.1.4 The Black and Scholes Formula

 4.2 A Useful Technique: Change of Measure

4.2.1 What is a Change of the Underlying Measure?

4.2.2 An Interpretation of the Black-Scholes Formula by Change of Measure

Appendix

 A1 Modes of Convergence

 A2 Inequalities

 A3 Non-Differentiability and Unbounded Variation of Brownian Sample Paths

 A4 Proof of the Existence of the General Ito Stochastic Integral

 A5 The Radon-Nikodym Theorem

AoProof of the Existence and Uniqueness of the Conditional Expectation

Bibliography

Index

List of Abbreviations and Symbols