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Continuous Parameter Markov Processes and Stochastic Differential Equations
Author
Title
Continuous Parameter Markov Processes and Stochastic Differential Equations [electronic resource] / by Rabi Bhattacharya, Edward C. Waymire.
ISBN
9783031332968
Edition
1st ed. 2023.
Publication
Cham : Springer International Publishing : Imprint: Springer, 2023.
Physical Description
1 online resource (XV, 506 p.) 4 illus.
Local Notes
Access is available to the Yale community.
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Summary
This graduate text presents the elegant and profound theory of continuous parameter Markov processes and many of its applications. The authors focus on developing context and intuition before formalizing the theory of each topic, illustrated with examples. After a review of some background material, the reader is introduced to semigroup theory, including the Hille-Yosida Theorem, used to construct continuous parameter Markov processes. Illustrated with examples, it is a cornerstone of Feller's seminal theory of the most general one-dimensional diffusions studied in a later chapter. This is followed by two chapters with probabilistic constructions of jump Markov processes, and processes with independent increments, or Lévy processes. The greater part of the book is devoted to Itô's fascinating theory of stochastic differential equations, and to the study of asymptotic properties of diffusions in all dimensions, such as explosion, transience, recurrence, existence of steady states, and the speed of convergence to equilibrium. A broadly applicable functional central limit theorem for ergodic Markov processes is presented with important examples. Intimate connections between diffusions and linear second order elliptic and parabolic partial differential equations are laid out in two chapters, and are used for computational purposes. Among Special Topics chapters, two study anomalous diffusions: one on skew Brownian motion, and the other on an intriguing multi-phase homogenization of solute transport in porous media.
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Springer ENIN.
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Format
Books / Online
Language
English
Added to Catalog
November 20, 2023
Series
Graduate Texts in Mathematics, 299
Contents
1. A review of Martingaels, stopping times and the Markov property
2. Semigroup theory and Markov processes.-3. Regularity of Markov process sample paths
4. Continuous parameter jump Markov processes
5. Processes with independent increments
6. The stochastic integral
7. Construction of difficusions as solutions of stochastic differential equations
8. Itô's Lemma
9. Cameron-Martin-Girsanov theorem
10. Support of nonsingular diffusions
11. Transience and recurrence of multidimensional diffusions
12. Criteria for explosion
13. Absorption, reflection and other transformations of Markov processes
14. The speed of convergence to equilibrium of discrete parameter Markov processes and Diffusions
15. Probabilistic representation of solutions to certain PDEs
16. Probabilistic solution of the classical Dirichlet problem
17. The functional Central Limit Theorem for ergodic Markov processes
18. Asymptotic stability for singular diffusions
19. Stochastic integrals with L2-Martingales
20. Local time for Brownian motion
21. Construction of one dimensional diffusions by Semigroups
22. Eigenfunction expansions of transition probabilities for one-dimensional diffusions
23. Special Topic: The Martingale Problem
24. Special topic: multiphase homogenization for transport in periodic media
25. Special topic: skew random walk and skew Brownian motion
26. Special topic: piecewise deterministic Markov processes in population biology
A. The Hille-Yosida theorem and closed graph theorem
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