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Elementary Stochastic Calculus.Thomas Mikosch
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. Thebook 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 It6 calculus and/or stochastic finance.
Reader Guidelines
1 Preliminaries
1.1 Basic Concepts from 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 Tile Interpretation of a Martingale as a FaiI: 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 It6 Integral
2.2.1 A Motivating Example
2.2.2 The It6 Stochastic Integral for Simple Processes
2.2.3 The General It6 Stochastic Integral
2.3 The It6 Lemma
2.3.1 The Classical Chain Rule of Differentiation
2.3.2 A Simple Version of the It6 Lemma
2.3.3 Extended Versions of the It6 Lemma
2.4 The Stratonovich and Other Integrals
3 Stochastic Differential Equations
3.1 Deterministic Differential Equations
3.2 It6 Stochastic Differential Equations
3.2.1 What is a Stochastic Differential Equation?
3.2.2 Solving It6 Stochastic Differential Equations by the It6 Lemma
3.2.3 Solving It6 Differential Equations via Stratonovich Cal-culus
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 tile Underlying Measure?
4.2.2 An Interpretation of the Black-Scholes Formula by Chan-ge of Measure
Appendix
A1 Modes of Convergence
A2 Inequalities
……
Bibliography
Index
List of Abbreviations and Symbols
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