Discretization of processes [electronic resource] / Jean Jacod, Philip Protter.
In applications, and especially in mathematical finance, random time-dependent events are often modeled as stochastic processes. Assumptions are made about the structure of such processes, and serious researchers will want to justify those assumptions through the use of data. As statisticians are wo...
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| Format: | Electronic eBook |
| Language: | English |
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Heidelberg ; New York :
Springer-Verlag Berlin Heidelberg,
©2012.
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| Series: | Stochastic modelling and applied probability ;
67. |
| Subjects: |
MARC
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| 100 | 1 | |a Jacod, Jean. |0 http://id.loc.gov/authorities/names/n80112452 |1 http://isni.org/isni/0000000116282311. | |
| 245 | 1 | 0 | |a Discretization of processes |h [electronic resource] / |c Jean Jacod, Philip Protter. |
| 260 | |a Heidelberg ; |a New York : |b Springer-Verlag Berlin Heidelberg, |c ©2012. | ||
| 300 | |a 1 online resource (xiv, 596 pages) | ||
| 336 | |a text |b txt |2 rdacontent. | ||
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| 490 | 1 | |a Stochastic modelling and applied probability, |x 0172-4568 ; |v 67. | |
| 504 | |a Includes bibliographical references and index. | ||
| 505 | 0 | |6 880-01 |a pt. 1. Introduction and preliminary material -- pt. 2. The basic results -- pt. 3. More laws of large numbers -- pt. 4. Extensions of the central limit theorems -- pt. 5. Various extensions. | |
| 520 | |a In applications, and especially in mathematical finance, random time-dependent events are often modeled as stochastic processes. Assumptions are made about the structure of such processes, and serious researchers will want to justify those assumptions through the use of data. As statisticians are wont to say, "In God we trust; all others must bring data." This book establishes the theory of how to go about estimating not just scalar parameters about a proposed model, but also the underlying structure of the model itself. Classic statistical tools are used: the law of large numbers, and the central limit theorem. Researchers have recently developed creative and original methods to use these tools in sophisticated (but highly technical) ways to reveal new details about the underlying structure. For the first time in book form, the authors present these latest techniques, based on research from the last 10 years. They include new findings. This book will be of special interest to researchers, combining the theory of mathematical finance with its investigation using market data, and it will also prove to be useful in a broad range of applications, such as to mathematical biology, chemical engineering, and physics. | ||
| 650 | 0 | |a Stochastic analysis. |0 http://id.loc.gov/authorities/subjects/sh85128175. | |
| 650 | 7 | |a Stochastic analysis. |2 fast |0 (OCoLC)fst01133499. | |
| 655 | 7 | |a Statistics. |2 fast |0 (OCoLC)fst01423727. | |
| 700 | 1 | |a Protter, Philip E. |0 http://id.loc.gov/authorities/names/n87892331 |1 http://isni.org/isni/0000000109351083. | |
| 776 | 0 | 8 | |i Print version: |z 9786613451521. |
| 830 | 0 | |a Stochastic modelling and applied probability ; |v 67. |0 http://id.loc.gov/authorities/names/no2005026362. | |
| 856 | 4 | 0 | |u https://colorado.idm.oclc.org/login?url=http://link.springer.com/10.1007/978-3-642-24127-7 |z Full Text (via Springer) |
| 880 | 0 | |6 505-01/(S |g Machine generated contents note: |g 1. |t Introduction -- |g 1.1. |t Content and Organization of the Book -- |g 1.2. |t When X is a Brownian Motion -- |g 1.2.1. |t Normalized Functionals V'n([ƒ], X) -- |g 1.2.2. |t Non-normalized Functionals Vn([ƒ], X) -- |g 1.3. |t When X is a Brownian Motion Plus Drift -- |g 1.3.1. |t Normalized Functionals V'n([ƒ], X) -- |g 1.3.2. |t Non-normalized Functionals Vn([ƒ], X) -- |g 1.4. |t When X is a Brownian Motion Plus Drift Plus a Compound Poisson Process -- |g 1.4.1. |t Law of Large Numbers -- |g 1.4.2. |t Central Limit Theorem -- |g 2. |t Some Prerequisites -- |g 2.1. |t Semimartingales -- |g 2.1.1. |t First Decompositions and the Basic Properties of a Semimartingale -- |g 2.1.2. |t Second Decomposition and Characteristics of a Semimartingale -- |g 2.1.3. |t Fundamental Example: Levy Processes -- |g 2.1.4. |t Itô Semimartingales -- |g 2.1.5. |t Some Estimates for Itô Semimartingales -- |g 2.1.6. |t Estimates for Bigger Filtrations -- |g 2.1.7. |t Lenglart Domination Property -- |g 2.2. |t Limit Theorems -- |g 2.2.1. |t Stable Convergence in Law -- |g 2.2.2. |t Convergence for Processes -- |g 2.2.3. |t Criteria for Convergence of Processes -- |g 2.2.4. |t Triangular Arrays: Asymptotic Negligibility -- |g 2.2.5. |t Convergence in Law of Triangular Arrays -- |t Bibliographical Notes -- |g 3. |t Laws of Large Numbers: The Basic Results -- |g 3.1. |t Discretization Schemes -- |g 3.2. |t Semimartingales with p-Summable Jumps -- |g 3.3. |t Law of Large Numbers Without Normalization -- |g 3.3.1. |t Results -- |g 3.3.2. |t Proofs -- |g 3.4. |t Law of Large Numbers with Normalization -- |g 3.4.1. |t Preliminary Comments -- |g 3.4.2. |t Results -- |g 3.4.3. |t Proofs -- |g 3.5. |t Applications -- |g 3.5.1. |t Estimation of the Volatility -- |g 3.5.2. |t Detection of Jumps -- |t Bibliographical Notes -- |g 4. |t Central Limit Theorems: Technical Tools -- |g 4.1. |t Processes with F-Conditionally Independent Increments -- |g 4.1.1. |t Continuous Case -- |g 4.1.2. |t Discontinuous Case -- |g 4.1.3. |t Mixed Case -- |g 4.2. |t Stable Convergence Result in the Continuous Case -- |g 4.3. |t Stable Convergence Result in the Discontinuous Case -- |g 4.4. |t Application to Itô Semimartingales -- |g 4.4.1. |t Localization Procedure -- |g 4.4.2. |t Stable Convergence for Itô Semimartingales -- |g 5. |t Central Limit Theorems: The Basic Results -- |g 5.1. |t Central Limit Theorem for Functionals Without Normalization -- |g 5.1.1. |t Central Limit Theorem, Without Normalization -- |g 5.1.2. |t Proof of the Central Limit Theorem, Without Normalization -- |g 5.2. |t Central Limit Theorem for Normalized Functionals: Centering with Conditional Expectations -- |g 5.2.1. |t Statement of the Results -- |g 5.2.2. |t Proof -- |g 5.3. |t Central Limit Theorem for the Processes V'n([ƒ], X) -- |g 5.3.1. |t Assumptions and Results -- |g 5.3.2. |t Localization and Elimination of Jumps -- |g 5.3.3. |t Proof of the Central Limit Theorem for V'n([ƒ], X) -- |g 5.4. |t Central Limit Theorem for Quadratic Variation -- |g 5.5. |t Joint Central Limit Theorem -- |g 5.6. |t Applications -- |g 5.6.1. |t Estimation of the Volatility -- |g 5.6.2. |t Detection of Jumps -- |g 5.6.3. |t Euler Schemes for Stochastic Differential Equations -- |t Bibliographical Notes -- |g 6. |t Integrated Discretization Error -- |g 6.1. |t Statements of the Results -- |g 6.2. |t Preliminaries -- |g 6.2.1. |t Application of Ito's Formula -- |g 6.2.2. |t Reduction of the Problem -- |g 6.3. |t Proof of the Theorems -- |g 6.3.1. |t Proof of Theorem 6.1.2 -- |g 6.3.2. |t Proof of Theorem 6.1.3 -- |g 6.3,3. |t Proof of Theorem 6.1.4 -- |g 6.3.4. |t Proof of Theorem 6.1.8 -- |g 7. |t First Extension: Random Weights -- |g 7.1. |t Introduction -- |g 7.2. |t Laws of Large Numbers for V'n (F, X) -- |g 7.3. |t Laws of Large Numbers for Vn(F, X) -- |g 7.4. |t Application to Some Parametric Statistical Problems -- |g 8. |t Second Extension: Functions of Several Increments -- |g 8.1. |t Introduction -- |g 8.2. |t Law of Large Numbers for Vn (F, X) and Vn (F, X) -- |g 8.3. |t Law of Large Numbers for Vn(&Φ, kn, X) -- |g 8.4. |t LLN for V'n(F, X), V'n(F, X) and V'n(Φ, kn, X) -- |g 8.4.1. |t Results -- |g 8.4.2. |t Proofs -- |g 8.5. |t Applications to Volatility -- |g 9. |t Third Extension: Truncated Functionals -- |g 9.1. |t Approximation for Jumps -- |g 9.2. |t Approximation for the Continuous Part of X -- |g 9.3. |t Local Approximation for the Continuous Part of X: Part I -- |g 9.4. |t From Local Approximation to Global Approximation -- |g 9.5. |t Local Approximation for the Continuous Part of X: Part II -- |g 9.6. |t Applications to Volatility -- |g 10. |t Central Limit Theorem for Random Weights -- |g 10.1. |t Functionals of Non-normalized Increments-Part I -- |g 10.2. |t Functionals of Non-normalized Increments-Part II -- |g 10.3. |t Functionals of Normalized Increments -- |g 10.4. |t Application to Parametric Estimation -- |t Bibliographical Notes -- |g 11. |t Central Limit Theorem for Functions of a Finite Number of Increments -- |g 11.1. |t Functionals of Non-normalized Increments -- |g 11.1.1. |t Results -- |g 11.1.2. |t Auxiliary Stable Convergence -- |g 11.1.3. |t Proof of Theorem 11.1.2 -- |g 11.2. |t Functionals of Normalized Increments -- |g 11.2.1. |t Results -- |g 11.2.2. |t Elimination of Jumps -- |g 11.2.3. |t Preliminaries for the Continuous Case -- |g 11.2.4. |t Processes Yn and yn -- |g 11.2.5. |t Proof of Lemma 11.2.7 -- |g 11.3. |t Joint Central Limit Theorems -- |g 11.4. |t Applications -- |g 11.4.1. |t Multipower Variations and Volatility -- |g 11.4.2. |t Sums of Powers of Jumps -- |g 11.4.3. |t Detection of Jumps -- |t Bibliographical Notes -- |g 12. |t Central Limit Theorem for Functions of an Increasing Number of Increments -- |g 12.1. |t Functionals of Non-normalized Increments -- |g 12.1.1. |t Results -- |g 12.1.2. |t Auxiliary Stable Convergence Result -- |g 12.1.3. |t Proof of Theorem 12.1.2 -- |g 12.2. |t Functionals of Normalized Increments -- |g 12.2.1. |t Results -- |g 12.2.2. |t Preliminaries for the Proof -- |g 12.2.3. |t Proof of Lemma 12.2.4 -- |g 12.2.4. |t Block Splitting -- |g 12.2.5. |t Proof of Lemma 12.2.3 -- |g 13. |t Central Limit Theorem for Truncated Functionals -- |g 13.1. |t Central Limit Theorem for Approximating the Jumps -- |g 13.2. |t Central Limit Theorem for Approximating the Continuous Part -- |g 13.2.1. |t Results -- |g 13.2.2. |t Proofs -- |g 13.3. |t Central Limit Theorem for the Local Approximation of the Continuous Part of X -- |g 13.3.1. |t Statements of Results -- |g 13.3.2. |t Elimination of the Jumps and of the Truncation -- |g 13.3.3. |t Scheme of the Proof in the Continuous Case -- |g 13.3.4. |t Proof of Lemma 13.3.12 -- |g 13.3.5. |t Proof of Lemma 13.3.13 -- |g 13.3.6. |t Proof of Theorem 13.3.8 -- |g 13.4. |t Another Central Limit Theorem Using Approximations of the Spot Volatility -- |g 13.4.1. |t Statements of Results -- |g 13.4.2. |t Proofs -- |g 13.5. |t Application to Volatility -- |t Bibliographical Notes -- |g 14. |t Irregular Discretization Schemes -- |g 14.1. |t Restricted Discretization Schemes -- |g 14.2. |t Law of Large Numbers for Normalized Functionals -- |g 14.3. |t Central Limit Theorem for Normalized Functionals -- |g 14.3.1. |t Results -- |g 14.3.2. |t Preliminaries -- |g 14.3.3. |t Scheme of the Proof when X is Continuous -- |g 14.3.4. |t Proof of Lemma 14.3.4 -- |g 14.3.5. |t Proof of Lemma 14.3.5 -- |g 14.4. |t Application to Volatility -- |t Bibliographical Notes -- |g 15. |t Higher Order Limit Theorems -- |g 15.1. |t Examples of Degenerate Situations -- |g 15.2. |t Functionals of Non-normalized Increments -- |g 15.3. |t Applications -- |t Bibliographical Notes -- |g 16. |t Semimartingales Contaminated by Noise -- |g 16.1. |t Structure of the Noise and the Pre-averaging Scheme -- |g 16.1.1. |t Structure of the Noise -- |g 16.1.2. |t Pre-averaging Scheme -- |g 16.2. |t Law of Large Numbers for General (Noisy) Semimartingales -- |g 16.3. |t Central Limit Theorem for Functionals of Non-normalized Increments -- |g 16.3.1. |t Results -- |g 16.3.2. |t Local Stable Convergence Result -- |g 16.3.3. |t Global Stable Convergence Result -- |g 16.3.4. |t Proof of Theorem 16.3.1 -- |g 16.4. |t Laws of Large Numbers for Normalized Functionals and Truncated Functionals -- |g 16.4.1. |t Statement of Results -- |g 16.4.2. |t Proofs -- |g 16.5. |t Laws of Large Numbers and Central Limit Theorems for Integral Power Functionals -- |g 16.5.1. |t Laws of Large Numbers -- |g 16.5.2. |t Central Limit Theorems: The Results. | |
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