1 Introduction 1.1 Stochastic Analogs of Classical Differential Equations 1.2 Filtering Problems 1.3 Stochastic Approach to Deterministic Boundary Value Problems 1.4 Optimal Stopping 1.5 Stochastic Control 1.6 Mathematical Finance 2 Some Mathematical Preliminaries 2.1 Probability Spaces, Random Variables and Stochastic Processes 2.2 An Important Example: Brownian Motion Exercises 3 It? Integrals 3.1 Construction of the It? Integral 3.2 Some properties of the It? integral 3.3 Extensions of the It? integral Exercises 4 The It? Formula and the Martingale Representation Theorem 4.1 The 1-dimensional It? formula 4.2 The Multi-dimensional It? Formula 4.3 The Martingale Representation Theorem Exercises 5 Stochastic Differential Equations 5.1 Examples and Some Solution Methods 5.2 An Existence and Uniqueness Result 5.3 Weak and Strong Solutions Exercises 6 The Filtering Problem 6.1 Introduction 6.2 The 1-Dimensional Linear Filtering Problem 6.3 The Multidimensional Linear Filtering Problem Exercises 7 Diffusions: Basic Properties 7.1 The Markov Property 7.2 The Strong Markov Property 7.3 The Generator of an It? Diffusion 7.4 The Dynkin Formula 7.5 The Characteristic Operator Exercises 8 Other Topics in Diffusion Theory 8.1 Kolmogorov's Backward Equation.The Resolvent 8.2 The Feynman-Kac Formula. Killing 8.3 The Martingale Problem 8.4 When is an It? Process a Diffusion 8.5 Random Time Change 8.6 The Girsanov Theorem Exercises 9 Applications to Boundary Value Problems 9.1 The Combined Dirichlet-Poisson Problem.Uniqueness 9.2 The Dirichlet Problem. Regular Points 9.3 The Poisson Problem
Exercises 10 Application to Optimal Stopping 10.1 The Time-Homogeneous Case 10.2 The Time-Inhomogeneous Case 10.3 Optimal Stopping Problems Involving an Integral 10.4 Connection with Variational Inequalities Exercises 11 Application to Stochastic Control 11.1 Statement of the Problem 11.2 The Hamilton-Jacobi-Bellman Equation 11.3 Stochastic control problems with terminal conditions Exercises 12 Application to Mathematical Finance 12.1 Market, portfolio and arbitrage 12.2 Attainability and Completeness 12.3 Option Pricing Exercises Appendix A: Normal Random Variables Appendix B: Conditional Expectation Appendix C: Uniform Integrability and Martingale Convergence Appendix D: An Approximation Result Solutions and Additional Hints to Some of the Exercises References List of Frequently Used Notation and Symbols Index
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