Mathematical finance requires the use of advanced mathematical techniques drawn from the theory of probability, stochastic processes and stochastic differential equations. These areas are generally introduced and developed at an abstract level, making it problematic when applying these techniques to practical issues in finance. Problems and Solutions in Mathematical Finance Volume I: Stochastic Calculus is the first of a four-volume set of books focusing on problems and solutions in mathematical finance. This volume introduces the reader to the basic stochastic calculus concepts required for the study of this important subject, providing a large number of worked examples which enable the reader to build the necessary foundation for more practical orientated problems in the later volumes. Through this application and by working through the numerous examples, the reader will properly understand and appreciate the fundamentals that underpin mathematical finance. Written mainly for students, industry practitioners and those involved in teaching in this field of study, Stochastic Calculus provides a valuable reference book to complement one s further understanding of mathematical finance.

Mathematical finance requires the use of advanced mathematicaltechniques drawn from the theory of probability, stochasticprocesses and stochastic differential equations. These areas aregenerally introduced and developed at an abstract level, making itproblematic when applying these techniques to practical issues infinance.Problems and Solutions in Mathematical Finance Volume I:Stochastic Calculus is the first of a four-volume set ofbooks focusing on problems and solutions in mathematicalfinance.This volume introduces the reader to the basic stochasticcalculus concepts required for the study of this important subjectproviding a large number of worked examples which enable the readerto build the necessary foundation for more practical orientatedproblems in the later volumes. Through this application and byworking through the numerous examples, the reader will properlyunderstand and appreciate the fundamentals that underpinmathematical finance.Written mainly for students, industry practitioners and thoseinvolved in teaching in this field of study, StochasticCalculus provides a valuable reference book to complementone's further understanding of mathematical finance.

Preface ixPrologue xiAbout the Authors xv1 General Probability Theory 11.1 Introduction 11.2 Problems and Solutions 41.2.1 Probability Spaces 41.2.2 Discrete and Continuous Random Variables 111.2.3 Properties of Expectations 412 Wiener Process 512.1 Introduction 512.2 Problems and Solutions 552.2.1 Basic Properties 552.2.2 Markov Property 682.2.3 Martingale Property 712.2.4 First Passage Time 762.2.5 Reflection Principle 842.2.6 Quadratic Variation 893 Stochastic Differential Equations 953.1 Introduction 953.2 Problems and Solutions 1023.2.1 Ito Calculus 1023.2.2 One-Dimensional Diffusion Process 1233.2.3 Multi-Dimensional Diffusion Process 1554 Change of Measure 1854.1 Introduction 1854.2 Problems and Solutions 1924.2.1 Martingale Representation Theorem 1924.2.2 Girsanov's Theorem 1944.2.3 Risk-Neutral Measure 2215 Poisson Process 2435.1 Introduction 2435.2 Problems and Solutions 2515.2.1 Properties of Poisson Process 2515.2.2 Jump Diffusion Process 2815.2.3 Girsanov's Theorem for Jump Processes 2985.2.4 Risk-Neutral Measure for Jump Processes 322Appendix A Mathematics Formulae 331Appendix B Probability Theory Formulae 341Appendix C Differential Equations Formulae 357Bibliography 365Notation 369Index 373