Mathematical models of financial markets rely in fundamental ways on the concepts and tools of modern probability theory. This book provides a concise but rigorous account of the probabilistic ideas and techniques most commonly used in such models. The treatment is self-contained, requiring only calculus and linear algebra as pre-requisites, and complete proofs are given – some longer constructions and proofs are deferred to the ends of chapters to ensure the smooth flow of key ideas.

New concepts are motivated through examples drawn from finance. The selection and ordering of the material are strongly guided by the applications we have in mind. Many of these applications appear more fully in later volumes of the ‘Mastering Mathematical Finance’ series, including [SCF], [BSM] and [NMFC]. This volume provides the essential mathematical background of the inancial models described in detail there.

In adding to the extensive literature on probability theory we have not sought to provide a comprehensive treatment of the mathematical theory and its manifold applications. We focus instead on the more limited objective of writing a fully rigorous, yet concise and accessible, account of the basic concepts underlying widely used market models. The book should be read in conjunction with its partner volume [SCF], which describes the properties of stochastic processes used in these models.