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.

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

Autorentext

Andrea Pascucci ist Professor für Wahrscheinlichkeitstheorie und mathematische Statistik an der Alma Mater Studiorum – Universität Bologna. Seine Forschungsaktivitäten umfassen verschiedene Aspekte der Theorie stochastischer Differentialgleichungen für Diffusions- und Sprungprozesse, degenerierte partielle Differentialgleichungen und deren Anwendungen in der mathematischen Finanzwirtschaft. Er hat sechs Bücher und über 80 wissenschaftliche Artikel zu folgenden Themen verfasst: lineare und nichtlineare Kolmogorov-Fokker-Planck-Gleichungen; Regularität und asymptotische Abschätzungen von Übergangsdichten für mehrdimensionale Diffusions- und Sprungprozesse; Freie Randwertprobleme, optimale Stopp-Probleme und Anwendungen auf amerikanische Finanzderivate; Asiatische Optionen und Volatilitätsmodelle. Er wurde als Referent zu mehr als 40 internationalen Konferenzen eingeladen. Er ist Herausgeber des Journal of Computational Finance und Leiter eines Postgraduierten-Programms für Mathematische Finanzwirtschaft an der Universität Bologna.

Inhalt

1 Stochastic processes.- 2 Markov processes.- 3 Continuous processes.- 4 Brownian motion.- 5 Poisson process.- 6 Stopping times.- 7 Strong Markov property.- 8 Continuous martingales.- 9 Theory of variation.- 10 Stochastic integral.- 11 Itô's formula.- 12 Multidimensional stochastic calculus.- 13 Change of measure and martingale representation.- 14 Stochastic differential equations.- 15 Feynman-Kac formulas.- 16 Linear stochastic equations.- 17 Strong solutions.- 18 Weak solutions.- 19 Complements.-20 A primer on parabolic PDEs.