Continuous Path Processes -- Continuous-Path Random Processes: Mathematical Prerequisites -- Basic Concepts and Examples in Finance -- Hitting Times: A Mix of Mathematics and Finance -- Complements on Brownian Motion -- Complements on Continuous Path Processes -- A Special Family of Diffusions: Bessel Processes -- Jump Processes -- Default Risk: An Enlargement of Filtration Approach -- Poisson Processes and Ruin Theory -- General Processes: Mathematical Facts -- Mixed Processes -- Lévy Processes.

Mathematical finance has grown into a huge area of research which requires a lot of care and a large number of sophisticated mathematical tools. The subject draws upon quite difficult results from the theory of stochastic processes, stochastic calculus and differential equations, among others, which can be daunting for the beginning researcher. This book simultaneously introduces the financial methodology and the relevant mathematical tools in a style that is mathematically rigorous and yet accessible to practitioners and mathematicians alike. It interlaces financial concepts such as arbitrage opportunities, admissible strategies, contingent claims, option pricing and default risk with the mathematical theory of Brownian motion, diffusion processes, and Lévy processes. The authors proceed by successive generalisations with increasing complexity assuming some basic knowledge of probability theory. The first half of the book is devoted to continuous path processes whereas the second half deals with discontinuous processes. The extensive bibliography comprises a wealth of important references and the author index enables readers quickly to locate where the reference is cited within the book, making this volume an invaluable tool both for students and for those at the forefront of research and practice.