Mathematical Finance, 1st ed. 2019 Springer Finance Series

Taking continuous-time stochastic processes allowing for jumps as its starting and focal point, this book provides an accessible introduction to the stochastic calculus and control of semimartingales and explains the basic concepts of Mathematical Finance such as arbitrage theory, hedging, valuation principles, portfolio choice, and term structure modelling. It bridges thegap between introductory texts and the advanced literature in the field.

Most textbooks on the subject are limited to diffusion-type models which cannot easily account for sudden price movements. Such abrupt changes, however, can often be observed in real markets. At the same time, purely discontinuous processes lead to a much wider variety of flexible and tractable models. This explains why processes with jumps have become an established tool in the statistics and mathematics of finance.

Graduate students, researchers as well as practitioners will benefit from this monograph. 

Part I.- Stochastic Calculus.- Overview.- Discrete Stochastic Calculus.- Lévy Processes.- Stochastic Integration.- Semimartingale Characteristics.- Markov Processes.- Affine and Polynomial Processes.- Optimal Control.- Mathematical Finance.- Overview and Notation.- Equity models.- Markets, Strategies, Arbitrage.- Optimal Investment.- Arbitrage-Based Valuation and Hedging of Derivatives.- Mean-Variance Hedging.- Utility-Based Valuation and Hedging of Derivatives.- Interest Rate Models.

Ernst Eberlein is professor emeritus at the University of Freiburg. After studying mathematics and physics at the universities of Erlangen and Paris, he received a Dr. rer. nat. at the University of Erlangen-Nürnberg and his habilitation in mathematics from ETH Zürich. For a period of ten years he served as Executive Secretary of the Bachelier Finance Society. From 2006 to 2013 he acted as co-editor of the journal Mathematical Finance.

Jan Kallsen is professor of mathematics at Kiel University. Having studied Mathematics and Physics in Kiel, Freiburg, Boston and Vienna, he received a Dr. rer. nat. and his habilitation from the University of Freiburg. Before coming to Kiel he held a position as professor of Mathematical Finance at the Technical University of Munich. 

Provides a gentle introduction to the calculus and control for stochastic processes with jumps

Covers Lévy and affine processes as well as their applications in financial modelling

Compares and explains the rationale behind different valuation and hedging concepts