Applied Stochastic Analysis
Seiten
2024
American Mathematical Society (Verlag)
9781470478391 (ISBN)
American Mathematical Society (Verlag)
9781470478391 (ISBN)
Stochastic analysis tackles randomness in physical systems by blending simulation methods with hands-on problem-solving strategies. It examines key concepts like Markov chains, Gaussian processes, Ito calculus and stochastic differential equations, building intuition for modeling evolving, time-dependent phenomena.
This textbook introduces the major ideas of stochastic analysis with a view to modeling or simulating systems involving randomness. Suitable for students and researchers in applied mathematics and related disciplines, this book prepares readers to solve concrete problems arising in physically motivated models. The author's practical approach avoids measure theory while retaining rigor for cases where it helps build techniques or intuition. Topics covered include Markov chains (discrete and continuous), Gaussian processes, Ito calculus, and stochastic differential equations and their associated PDEs. We ask questions such as: How does probability evolve? How do statistics evolve? How can we solve for time-dependent quantities such as first-passage times? How can we set up a model that includes fundamental principles such as time-reversibility (detailed balance)? How can we simulate a stochastic process numerically? Applied Stochastic Analysis invites readers to develop tools and insights for tackling physical systems involving randomness. Exercises accompany the text throughout, with frequent opportunities to implement simulation algorithms. A strong undergraduate background in linear algebra, probability, ODEs, and PDEs is assumed, along with the mathematical sophistication characteristic of a graduate student.
This textbook introduces the major ideas of stochastic analysis with a view to modeling or simulating systems involving randomness. Suitable for students and researchers in applied mathematics and related disciplines, this book prepares readers to solve concrete problems arising in physically motivated models. The author's practical approach avoids measure theory while retaining rigor for cases where it helps build techniques or intuition. Topics covered include Markov chains (discrete and continuous), Gaussian processes, Ito calculus, and stochastic differential equations and their associated PDEs. We ask questions such as: How does probability evolve? How do statistics evolve? How can we solve for time-dependent quantities such as first-passage times? How can we set up a model that includes fundamental principles such as time-reversibility (detailed balance)? How can we simulate a stochastic process numerically? Applied Stochastic Analysis invites readers to develop tools and insights for tackling physical systems involving randomness. Exercises accompany the text throughout, with frequent opportunities to implement simulation algorithms. A strong undergraduate background in linear algebra, probability, ODEs, and PDEs is assumed, along with the mathematical sophistication characteristic of a graduate student.
Miranda Holmes-Cerfon, University of British Columbia, Vancouver, BC, Canada
Introduction
Markov chains (I)
Markov chains (II): Detailed balance, and Markov chain Monte Carlo (MCMC)
Continuous-time Markov chains
Gaussian processes & stationary processes
Brownian motion
Stochastic integration
Stochastic differential equations
Numerically solvding SDEs
Forward and backward equations for SDEs
Some applicationis of the backward equation
Detailed balance, symmetry, and eigenfunction expansions
Asymptotic analysis of SDEs
Appendix
Bibliography
Index
| Erscheinungsdatum | 19.03.2025 |
|---|---|
| Reihe/Serie | Courant Lecture Notes |
| Verlagsort | Providence |
| Sprache | englisch |
| Maße | 178 x 254 mm |
| Themenwelt | Mathematik / Informatik ► Mathematik |
| ISBN-13 | 9781470478391 / 9781470478391 |
| Zustand | Neuware |
| Informationen gemäß Produktsicherheitsverordnung (GPSR) | |
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