Stochastic Differential Equations: An Introduction with Applications

Description

An introduction to the basic theory of stochastic calculus and its applications. Examples are given throughout the text, in order to motivate and illustrate the theory and show its importance for many applications in e.g. economics, biology and physics. The basic idea of the presentation is to start from some basic results (without proofs) of the easier cases and develop the theory from there, and to concentrate on the proofs of the easier case in order to quickly progress to the parts of the theory that are most important for the applications. For the 6th edition the author has added further exercises and, for the first time, solutions to many of the exercises are provided. Preface to the Sixth Corrected Printing of the Sixth Edition Preface to the Fifth Corrected Printing of the Sixth Edition Preface to the Fourth Corrected Printing of the Sixth Edition Preface to the Third Corrected Printing of the Sixth Edition Preface to the Sixth Edition Preface to Corrected Printing, Fifth Edition Preface to the Fifth Edition Preface to the Fourth Edition Preface to the Third Edition Preface to the Second Edition Preface to the First Edition Contents 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 StochasticProcesses 2.2 An Important Example: Brownian Motion Exercises 3 Itô Integrals 3.1 Construction of the Itô Integral 3.2 Some properties of the Itô integral 3.3 Extensions of the Itô integral Exercises 4 The Itô Formula and the Martingale Representation Theorem 4.1 The 1-dimensional Itô formula 4.2 The Multi-dimensional Itô 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 Itô 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 Itô 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 MartingaleConvergence Appendix D: An Approximation Result Solutions and Additional Hints to Some of the Exercises References List of Frequently Used Notation and Symbols Index