Preface

Summary

In this volume of the series ‘Mastering Mathematical Finance’ we develop the essential tools from stochastic calculus that will be needed in later volumes for the rigorous development of the Black-Scholes option pricing model and various of its extensions. Our motivation, and hence our choice of material, is again taken from the applications we have in mind: we develop only those parts of the theory that will be indispensable for the financial models discussed in this series. The Itô integral, with the Wiener process as its driving force, forms the heart of the text, with the Itô formula, developed in stages until we reach a sufficiently general setting, as the principal tool of our calculus.

The initial chapter sets the scene with an account of the basics of martingale theory in discrete time, and a brief introduction to Markov chains. The focus then shifts to continuous time, with a careful construction and development of the principal path, martingale and Markov properties of the Wiener process, followed by the construction of the Itô integral and discussion of its key properties. Itô processes are discussed next, and their quadratic variations are identified. Chapter 4 focuses on a complete proof of the Itô formula, which is often omitted in introductory texts, or presented as a by-product of more advanced treatments. The stringent boundedness assumptions required by an elementary treatment are removed by means of localisation, and the role of local martingales is emphasised.