The book provides an introduction to advanced topics in stochastic processes and related stochastic analysis, and combines them with a sound presentation of the fundamentals of financial mathematics. It is wide-ranging in content, while at the same time placing much emphasis on good readability, motivation, and explanation of the issues covered. 

Financial mathematical topics are first introduced in the context of discrete time processes and then transferred to continuous-time modelsThe basic construction of the stochastic integral and the associated martingale theory provide fundamental methods of the theory of stochastic processes for the construction of suitable stochastic models of financial mathematics, e.g. using stochastic differential equations.

Central results of stochastic analysis such as the Itô formula, Girsanov''s theorem and martingale representation theorems are of fundamental importance in financial mathematics, e.g. for the risk-neutral valuation formula (Black-Scholes formula) or the question of the hedgeability of options and the completeness of market models.

Chapters on the valuation of options in complete and incomplete markets and on the determination of optimal hedging strategies conclude the range of topics.