Stochastic Processes and Financial Mathematics

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This book introduces advanced topics in stochastic processes and stochastic analysis, integrating them with the fundamentals of financial mathematics. It emphasizes readability, motivation, and clear explanations while covering a wide range of content. Financial mathematics is first explored through discrete time processes before transitioning to continuous-time models. Key concepts such as the stochastic integral and martingale theory are essential for developing stochastic models in finance, particularly through stochastic differential equations. Important results in stochastic analysis, including the Itô formula, Girsanov's theorem, and martingale representation theorems, play a crucial role in financial mathematics, influencing risk-neutral valuation (e.g., the Black-Scholes formula) and the hedgeability of options in market models. The book also addresses option valuation in both complete and incomplete markets and optimal hedging strategies. It assumes advanced knowledge of probability theory, particularly in discrete-time processes (martingales, Markov chains) and continuous-time processes (Brownian motion, Lévy processes). This makes it suitable for advanced students and instructors. The text is a translation of the original German edition, enhanced with AI assistance and author revisions.