Overview
- Rigorous, comprehensive, and self-contained presentation
- Written clearly to make complex mathematics accessible
- Comprehensive overview of stochastic process theory with brief mentions of its most significant applications
Part of the book series: UNITEXT (UNITEXT, volume 166)
Part of the book sub series: La Matematica per il 3+2 (UNITEXTMAT)
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Keywords
- Stochastic differential equations
- Martingale
- Brownian motion
- Ito integral
- Markov process
- Stochastic calculus
- Ito formula
- Stochastic process
About this book
This book offers a modern approach to the theory of continuous-time stochastic processes and stochastic calculus. The content is treated rigorously, comprehensively, and independently. In the first part, the theory of Markov processes and martingales is introduced, with a focus on Brownian motion and the Poisson process. Subsequently, the theory of stochastic integration for continuous semimartingales was developed. A substantial portion is dedicated to stochastic differential equations, the main results of solvability and uniqueness in weak and strong sense, linear stochastic equations, and their relation to deterministic partial differential equations. Each chapter is accompanied by numerous examples. This text stems from over twenty years of teaching experience in stochastic processes and calculus within master's degrees in mathematics, quantitative finance, and postgraduate courses in mathematics for applications and mathematical finance at the University of Bologna. The book provides material for at least two semester-long courses in scientific studies (Mathematics, Physics, Engineering, Statistics, Economics, etc.) and aims to provide a solid background for those interested in the development of stochastic calculus theory and its applications. This text completes the journey started with the first volume of Probability Theory I - Random Variables and Distributions, through a selection of advanced classic topics in stochastic analysis.
Authors and Affiliations
About the author
Andrea Pascucci is a professor of Probability and Mathematical Statistics at the Alma Mater Studiorum – University of Bologna. His research activity encompasses various aspects of the theory of stochastic differential equations for diffusions and jump processes, degenerate partial differential equations, and their applications to mathematical finance. He has authored 6 books and over 80 scientific articles on the following topics: linear and nonlinear Kolmogorov-Fokker-Planck equations; regularity and asymptotic estimates of transition densities for multidimensional diffusions and jump processes; free boundary problems, optimal stopping, and applications to American-style financial derivatives; Asian options and volatility models. He has been invited as a speaker at more than 40 international conferences. He serves as an editor for the Journal of Computational Finance and is the director of a postgraduate program in Mathematical Finance at the University of Bologna.
Bibliographic Information
Book Title: Probability Theory II
Book Subtitle: Stochastic Calculus
Authors: Andrea Pascucci
Series Title: UNITEXT
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2025
Softcover ISBN: 978-3-031-63192-4Due: 20 October 2024
eBook ISBN: 978-3-031-63193-1Due: 20 October 2024
Series ISSN: 2038-5714
Series E-ISSN: 2532-3318
Edition Number: 1
Number of Illustrations: 4 b/w illustrations, 14 illustrations in colour
Additional Information: Translation from the Italian language edition: “Teoria della Probabilità. Processi e calcolo stocastico” by Andrea Pascucci, © Springer-Verlag Italia S.r.l., part of Springer Nature 2024.