| Description |
1 online resource (xv, 390 pages) : illustrations. |
| Physical Medium |
polychrome |
| Description |
text file |
| Series |
Cambridge series in statistical and probabilistic mathematics ; 33
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Cambridge series on statistical and probabilistic mathematics ; 33.
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| Summary |
"This comprehensive guide to stochastic processes gives a complete overview of the theory and addresses the most important applications. Pitched at a level accessible to beginning graduate students and researchers from applied disciplines, it is both a course book and a rich resource for individual readers. Subjects covered include Brownian motion, stochastic calculus, stochastic differential equations, Markov processes, weak convergence of processes and semigroup theory. Applications include the Black-Scholes formula for the pricing of derivatives in financial mathematics, the Kalman-Bucy filter used in the US space program and also theoretical applications to partial differential equations and analysis. Short, readable chapters aim for clarity rather than full generality. More than 350 exercises are included to help readers put their new-found knowledge to the test and to prepare them for tackling the research literature"-- Provided by publisher. |
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"In a first course on probability one typically works with a sequence of random variables X1,X2 ... For stochastic processes, instead of indexing the random variables by the non-negative integers, we index them by t G [0, oo) and we think of Xt as being the value at time t. The random variable could be the location of a particle on the real line, the strength of a signal, the price of a stock, and many other possibilities as well. We will also work with increasing families of s -fields {J-t}, known as filtrations. The s -field J-t is supposed to represent what we know up to time t. 1.1 Processes and s -fields Let (Q., J-, P) be a probability space. A real-valued stochastic process (or simply a process) is a map X from [0, oo) x Q. to the reals. We write Xt = Xt = X(t,?). We will impose stronger measurability conditions shortly, but for now we require that the random variables Xt be measurable with respect to J- for each t 0. A collection of s -fields J-t such that J-t C J- for each t and J-s C J-t if s t is called a filtration. Define J-t+ = P\e0J-t+e. A filtration is right continuous if J-t+ = J-t for all t 0"-- Provided by publisher. |
| Bibliography |
Includes bibliographical references and index. |
| Contents |
1. Basic notions -- 2. Brownian motion -- 3. Martingales -- 4. Markov properties of Brownian motion -- 5. The Poisson process -- 6. Construction of Brownian motion -- 7. Path properties of Brownian motion -- 8. The continuity of paths -- 9. Continuous semimartingales -- 10. Stochastic integrals -- 11. Itô's formula -- 12. Some applications of Itô's formula -- 13. The Girsanov theorem -- 14. Local times -- 15. Skorokhod embedding -- 16. The general theory of processes -- 17. Processes with jumps -- 18. Poisson point processes -- 19. Framework for Markov processes -- 20. Markov properties -- 21. Applications of the Markov properties -- 22. Transformations of Markov processes -- 23. Optimal stopping -- 24. Stochastic differential equations -- 25. Weak solutions of SDEs -- 26. The Ray-Knight theorems -- 27. Brownian excursions -- 28. Financial mathematics -- 29. Filtering -- 30. Convergence of probability measures -- 31. Skorokhod representation -- 32. The space C[0, 1] -- 33. Gaussian processes -- 34. The space D[0, 1] -- 35. Applications of weak convergence -- 36. Semigroups -- 37. Infinitesimal generators -- 38. Dirichlet forms -- 39. Markov processes and SDEs -- 40. Solving partial differential equations -- 41. One-dimensional diffusions -- 42. Lévy processes -- Appendices: A. Basic probability; B. Some results from analysis; C. Regular conditional probabilities; D. Kolmogorov extension theorem. |
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Machine generated contents note: 1. Basic notions -- 1.1. Processes and σ-fields -- 1.2. Laws and state spaces -- 2. Brownian motion -- 2.1. Definition and basic properties -- 3. Martingales -- 3.1. Definition and examples -- 3.2. Doob's inequalities -- 3.3. Stopping times -- 3.4. optional stopping theorem -- 3.5. Convergence and regularity -- 3.6. Some applications of martingales -- 4. Markov properties of Brownian motion -- 4.1. Markov properties -- 4.2. Applications -- 5. Poisson process -- 6. Construction of Brownian motion -- 6.1. Wiener's construction -- 6.2. Martingale methods -- 7. Path properties of Brownian motion -- 8. continuity of paths -- 9. Continuous semimartingales -- 9.1. Definitions -- 9.2. Square integrable martingales -- 9.3. Quadratic variation -- 9.4. Doob-Meyer decomposition -- 10. Stochastic integrals -- 10.1. Construction -- 10.2. Extensions -- 11. Ito's formula -- 12. Some applications of Ito's formula -- 12.1. Levy's theorem -- 12.2. Time changes of martingales -- 12.3. Quadratic variation -- 12.4. Martingale representation -- 12.5. Burkholder-Davis-Gundy inequalities -- 12.6. Stratonovich integrals -- 13. Girsanov theorem -- 13.1. Brownian motion case -- 13.2. example -- 14. Local times -- 14.1. Basic properties -- 14.2. Joint continuity of local times -- 14.3. Occupation times -- 15. Skorokhod embedding -- 15.1. Preliminaries -- 15.2. Construction of the embedding -- 15.3. Embedding random walks -- 16. general theory of processes -- 16.1. Predictable and optional processes -- 16.2. Hitting times -- 16.3. debut and section theorems -- 16.4. Projection theorems -- 16.5. More on predictability -- 16.6. Dual projection theorems -- 16.7. Doob-Meyer decomposition -- 16.8. Two inequalities -- 17. Processes with jumps -- 17.1. Decomposition of martingales -- 17.2. Stochastic integrals -- 17.3. Ito's formula -- 17.4. reduction theorem -- 17.5. Semimartingales -- 17.6. Exponential of a semimartingale -- 17.7. Girsanov theorem -- 18. Poisson point processes -- 19. Framework for Markov processes -- 19.1. Introduction -- 19.2. Definition of a Markov process -- 19.3. Transition probabilities -- 19.4. example -- 19.5. canonical process and shift operators -- 20. Markov properties -- 20.1. Enlarging the filtration -- 20.2. Markov property -- 20.3. Strong Markov property -- 21. Applications of the Markov properties -- 21.1. Recurrence and transience -- 21.2. Additive functionals -- 21.3. Continuity -- 21.4. Harmonic functions -- 22. Transformations of Markov processes -- 22.1. Killed processes -- 22.2. Conditioned processes -- 22.3. Time change -- 22.4. Last exit decompositions -- 23. Optimal stopping -- 23.1. Excessive functions -- 23.2. Solving the optimal stopping problem -- 24. Stochastic differential equations -- 24.1. Pathwise solutions of SDEs -- 24.2. One-dimensional SDEs -- 24.3. Examples of SDEs -- 25. Weak solutions of SDEs -- 26. Ray-Knight theorems -- 27. Brownian excursions -- 28. Financial mathematics -- 28.1. Finance models -- 28.2. Black-Scholes formula -- 28.3. fundamental theorem of finance -- 28.4. Stochastic control -- 29. Filtering -- 29.1. basic model -- 29.2. innovation process -- 29.3. Representation of FZ-martingales -- 29.4. filtering equation -- 29.5. Linear models -- 29.6. Kalman-Bucy filter -- 30. Convergence of probability measures -- 30.1. portmanteau theorem -- 30.2. Prohorov theorem -- 30.3. Metrics for weak convergence -- 31. Skorokhod representation -- 32. space C[0, 1] -- 32.1. Tightness -- 32.2. construction of Brownian motion -- 33. Gaussian processes -- 33.1. Reproducing kernel Hilbert spaces -- 33.2. Continuous Gaussian processes -- 34. space D[0, 1] -- 34.1. Metrics for D[0, 1] -- 34.2. Compactness and completeness -- 34.3. Aldous criterion -- 35. Applications of weak convergence -- 35.1. Donsker invariance principle -- 35.2. Brownian bridge -- 35.3. Empirical processes -- 36. Semigroups -- 36.1. Constructing the process -- 36.2. Examples -- 37. Infinitesimal generators -- 37.1. Semigroup properties -- 37.2. Hille-Yosida theorem -- 37.3. Nondivergence form elliptic operators -- 37.4. Generators of Levy processes -- 38. Dirichlet forms -- 38.1. Framework -- 38.2. Construction of the semigroup -- 38.3. Divergence form elliptic operators -- 39. Markov processes and SDEs -- 39.1. Markov properties -- 39.2. SDEs and PDEs -- 39.3. Martingale problems -- 40. Solving partial differential equations -- 40.1. Poisson's equation -- 40.2. Dirichlet problem -- 40.3. Cauchy problem -- 40.4. Schrodinger operators -- 41. One-dimensional diffusions -- 41.1. Regularity -- 41.2. Scale functions -- 41.3. Speed measures -- 41.4. uniqueness theorem -- 41.5. Time change -- 41.6. Examples -- 42. Levy processes -- 42.1. Examples -- 42.2. Construction of Levy processes -- 42.3. Representation of Levy processes -- Appendices -- A. Basic probability -- A.1. First notions -- A.2. Independence -- A.3. Convergence -- A.4. Uniform integrability -- A.5. Conditional expectation -- A.6. Stopping times -- A.7. Martingales -- A.8. Optional stopping -- A.9. Doob's inequalities -- A.10. Martingale convergence theorem -- A.11. Strong law of large numbers -- A.12. Weak convergence -- A.13. Characteristic functions -- A.14. Uniqueness and characteristic functions -- A.15. central limit theorem -- A.16. Gaussian random variables -- B. Some results from analysis -- B.1. monotone class theorem -- B.2. Schwartz class -- C. Regular conditional probabilities -- D. Kolmogorov extension theorem. |
| Local Note |
eBooks on EBSCOhost EBSCO eBook Subscription Academic Collection - North America |
| Subject |
Stochastic analysis.
|
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Stochastic analysis. |
| Genre/Form |
Electronic books.
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Electronic book.
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| Other Form: |
Print version: Bass, Richard F. Stochastic processes. Cambridge ; New York : Cambridge University Press, 2011 9781107008007 (DLC) 2011023024 (OCoLC)711048245 |
| ISBN |
9781139128452 (electronic book) |
|
1139128450 (electronic book) |
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1139115626 |
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9781139115629 |
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9780511997044 (electronic book) |
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0511997043 (electronic book) |
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9781139117791 |
|
1139117793 |
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9781107008007 |
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110700800X |
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9781139115629 |
| Standard No. |
9786613315007 |
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