Elementary Stochastic Calculus, with Fin

Reader Guidelines 1(4) Preliminaries 5(82) Basic Concepts from Probability Theory 6(17) Random Variables 6(8) Random Vectors 14(5) Independence and Dependence 19(4) Stochastic Processes 23(10) Brownian Motion 33(23) Defining Properties 33(7) Processes Derived from Brownian Motion 40(4) Simulation of Brownian Sample Paths 44(12) Conditional Expectation 56(21) Conditional Expectation under Discrete Condition 56(6) About σ-Fields 62(5) The General Conditional Expectation 67(3) Rules for the Calculation of Conditional Expectations 70(4) The Projection Property of Conditional Expectations 74(3) Martingales 77(10) Defining Properties 77(4) Examples 81(3) The Interpretation of a Martingale as a Fair Game 84(3) The Stochastic Integral 87(44) The Riemann and Riemann--Stieltjes Integrals 88(8) The Ordinary Riemann Integral 88(4) The Riemann--Stieltjes Integral 92(4) The Ito Integral 96(16) A Motivating Example 96(5) The Ito Stochastic Integral for Simple Processes 101(6) The General Ito Stochastic Integral 107(5) The Ito Lemma 112(11) The Classical Chain Rule of Differentiation 113(1) A Simple Version of the Ito Lemma 114(3) Extended Versions of the Ito Lemma 117(6) The Stratonovich and Other Integrals 123(8) Stochastic Differential Equations 131(36) Deterministic Differential Equations 132(2) Ito Stochastic Differential Equations 134(16) What is a Stochastic Differential Equation? 134(4) Solving Ito Stochastic Differential Equations by the Ito Lemma 138(7) Solving Ito Differential Equations via Stratonovich Calculus 145(5) The General Linear Differential Equation 150(7) Linear Equations with Additive Noise 150(3) Homogeneous Equations with Multiplicative Noise 153(2) The General Case 155(1) The Expectation and Variance Functions of the Solution 156(1) Numerical Solution 157(10) The Euler Approximation 158(4) The Milstein Approximation 162(5) Applications of Stochastic Calculus in Finance 167(18) The Black--Scholes Option Pricing Formula 168(8) A Short Excursion into Finance 168(2) What is an Option? 170(2) A Mathematical Formulation of the Option Pricing Problem 172(2) The Black and Scholes Formula 174(2) A Useful Technique: Change of Measure 176(9) What is a Change of the Underlying Measure? 176(4) An Interpretation of the Black--Scholes Formula by Change of Measure 180(5) Appendix 185(1) A1 Modes of Convergence 185(2) A2 Inequalities 187(1) A3 Non-Differentiability and Unbounded Variation of Brownian Sample Paths 188(2) A4 Proof of the Existence of the General Ito Stochastic Integral 190(3) A5 The Radon--Nikodym Theorem 193(1) A6 Proof of the Existence and Uniqueness of the Conditional Expectation 194(1) Bibliography 195(4) Index 199(10) List of Abbreviations and Symbols 209