Problems and Solutions in Mathematical Finance - Stochastic Calculus

Problems and Solutions in Mathematical Finance - Stochastic Calculus

TABLE OF CONTENTS

Preface ix

Prologue xi

About the Authors xv

1 General Probability Theory 1

1.1 Introduction 1

1.2 Problems and Solutions 4

1.2.1 Probability Spaces 4

1.2.2 Discrete and Continuous Random Variables 11

1.2.3 Properties of Expectations 41

2 Wiener Process 51

2.1 Introduction 51

2.2 Problems and Solutions 55

2.2.1 Basic Properties 55

2.2.2 Markov Property 68

2.2.3 Martingale Property 71

2.2.4 First Passage Time 76

2.2.5 Reflection Principle 84

2.2.6 Quadratic Variation 89

3 Stochastic Differential Equations 95

3.1 Introduction 95

3.2 Problems and Solutions 102

3.2.1 It¯o Calculus 102

3.2.2 One-Dimensional Diffusion Process 123

3.2.3 Multi-Dimensional Diffusion Process 155

4 Change of Measure 185

4.1 Introduction 185

4.2 Problems and Solutions 192

4.2.1 Martingale Representation Theorem 192

4.2.2 Girsanov’s Theorem 194

4.2.3 Risk-Neutral Measure 221

5 Poisson Process 243

5.1 Introduction 243

5.2 Problems and Solutions 251

5.2.1 Properties of Poisson Process 251

5.2.2 Jump Diffusion Process 281

5.2.3 Girsanov’s Theorem for Jump Processes 298

5.2.4 Risk-Neutral Measure for Jump Processes 322

Appendix A Mathematics Formulae 331

Appendix B Probability Theory Formulae 341

Appendix C Differential Equations Formulae 357

Bibliography 365

Notation 369

Index 373