I knew better.At that time.staft members of economics and mathematicsdepartments already discussed the use of the Black and Scholes option pricingformula；courses on stochastic finance were 0fiered at leading institutions suchas ETH Zfirich.Columbia and Stanford；and there Was a general agreementthat not only students and staft members of economics and mathematics de-partments、but also practitioners in financiai institutions should know moreabout this new topic.
　　Soon I realized that there Was not very much literature which could beused for teaching stochastic caiculus at a rather elementary level.I aln fullyaware of the fact that a combination of“elementary”and“stochastic calculus”is a contradiction iU itself Stochastic calculus requires advanced mathematicaitechniques；this theory cannot be fullv understood if one does not know aboutthe basics of measure theory，functional analysis and the theory of stochasticprocesses However.I strongly believe that an interested person who knowsabout elementary probability theory and who can handle the rules of inte-gration and difierentiation is able to understand the main ideas of stochasticcalculus.This is supported by my experience which I gained in courses foreconomics statistics and mathematics students at VUW Wellington and theDepartment of Mathematics in Groningen.I got the same impression as alecturer of crash courses on stochastic calculus at the Summer SchOOl.

　　Ten years ago 1 would not have dared to write a book like this：a non-rigorous treatment of a mathematician theorys admit that 1 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.　I started writing this book as lecture notes in 1992 when I prepared a course on stochastic calculus for the students of the Commerce Faculty at Victoria University Wellington （New Zealand）.Since I had failed in giving tutorials on portfolio theory and investment analysis. I Was expected to teach something
　　I knew better.At that time.staft members of economics and mathematicsdepartments already discussed the use of the Black and Scholes option pricingformula；courses on stochastic finance were 0fiered at leading institutions suchas ETH Zfirich.Columbia and Stanford；and there Was a general agreementthat not only students and staft members of economics and mathematics de-partments、but also practitioners in financiai institutions should know moreabout this new topic.
　　Soon I realized that there Was not very much literature which could beused for teaching stochastic caiculus at a rather elementary level.I aln fullyaware of the fact that a combination of“elementary”and“stochastic calculus”is a contradiction iU itself Stochastic calculus requires advanced mathematicaitechniques；this theory cannot be fullv understood if one does not know aboutthe basics of measure theory，functional analysis and the theory of stochasticprocesses However.I strongly believe that an interested person who knowsabout elementary probability theory and who can handle the rules of inte-gration and difierentiation is able to understand the main ideas of stochasticcalculus.This is supported by my experience which I gained in courses foreconomics statistics and mathematics students at VUW Wellington and theDepartment of Mathematics in Groningen.I got the same impression as alecturer of crash courses on stochastic calculus at the Summer SchOOl.

Reader Guidelines
1 Preliminaries
1.1 Basic Concepts flom 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