Mathematical finance is one of the most popular courses at universities and the development in this field is getting more and more sophisticated. Although the progress in mathematical finance has been substantial in recent years, the mathematical principles behind it have not changed in any fundamental way. This book will provide a thorough treatment of stochastic calculus and probability theory that underpin the theory of mathematical finance. The book will take a dual approach - to use stochastic calculus to develop partial differentiation equations for pricing options and also to construct probability measures via martingale theory so that option prices can be expressed as expectations. Each chapter will begin with an introduction of the fundamentals, and the essential definitions and explanations needed to solve the subsequent problems. For students and practitioners, as well as those involved in teaching mathematical finance and financial engineering, this book can be used as a supplementary text or as an independent self-study tool. Volume I Problems and Solutions on Mathematical Finance : Stochastic Calculus Table of Contents 1. General Probability and Statistical Theory 1.

1 Introduction 1.2 Problems and Solutions 1.2.1 Probability Spaces 1.2.2 Discrete and Continuous Random Variables 1.2.3 Properties of Expectations 2.

General Statistical Theory 2.1 Introduction 2.2 Problems and Solutions 2.2.1 Parameter Estimation 2.2.2 Hypotheses Testing 2.2.

3 Goodness of Fit Analysis 2.2.4 Regression Analysis 3. Wiener Process 3.1 Introduction 3.2 Problems and Solutions 3.2.1 Random Walks 3.

2.2 Examples of Wiener Process 3.2.3 Markov Property 3.2.4 Martingale Property 3.2.5 First Passage Time 3.

2.6 Reflection Principle 3.2.7 Quadratic Variation 4. Stochastic Differential Equations 4.1 Introduction 4.2 Problems and Solutions 4.2.

1 Ito Calculus 4.2.2 One-Dimension Diffusion Process 4.2.3 Multi-Dimensional Diffusion Process 5. Change of Measure 5.1 Introduction 5.2 Problems and Solutions 5.

2.1 Martingale Representation Theorem 5.2.2 Girsanov's Theorem 5.2.3 Risk Neutral Measure 6. Poisson Process 6.1 Introduction 6.

2 Problems and Solutions 6.2.1 Properties of Poisson Process 6.2.2 Jump Diffusion Process 6.2.3 Change of Measure Appendix A Mathematics Formulae Appendix B Probability Theory Formulae Appendix C Statistical Theory Formulae Appendix D Differential Equations Formulae.