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From Lévy-Type Processes to Parabolic SPDEs / / by Davar Khoshnevisan, René Schilling ; edited by Frederic Utzet, Lluis Quer-Sardanyons
| From Lévy-Type Processes to Parabolic SPDEs / / by Davar Khoshnevisan, René Schilling ; edited by Frederic Utzet, Lluis Quer-Sardanyons |
| Autore | Khoshnevisan Davar |
| Edizione | [1st ed. 2016.] |
| Pubbl/distr/stampa | Cham : , : Springer International Publishing : , : Imprint : Birkhäuser, , 2016 |
| Descrizione fisica | 1 online resource (VIII, 219 p.) |
| Disciplina | 519.2 |
| Collana | Advanced Courses in Mathematics - CRM Barcelona |
| Soggetto topico |
Probabilities
Differential equations, Partial Probability Theory and Stochastic Processes Partial Differential Equations |
| ISBN | 3-319-34120-0 |
| Formato | Materiale a stampa  |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Intro -- Contents -- Part I: An Introduction to Lévy and Feller Processes -- Preface -- Symbols and Notation -- Chapter 1: Orientation -- Chapter 2: Lévy Processes -- Chapter 3: Examples -- Chapter 4: On the Markov Property -- Chapter 5: A Digression: Semigroups -- Chapter 6: The Generator of a Lévy Process -- Chapter 7: Construction of Lévy Processes -- Chapter 8: Two Special Lévy Processes -- Chapter 9: Random Measures -- Chapter 10: A Digression: Stochastic Integrals -- Chapter 11: From Lévy to Feller Processes -- Chapter 12: Symbols and Semimartingales -- Chapter 13: Dénouement -- Appendix: Some Classical Results -- The Cauchy-Abel functional equation -- Characteristic functions and moments -- Vague and weak convergence of measures -- Convergence in distribution -- The predictable σ-algebra -- The structure of translation invariant operators -- Bibliography -- Part II: Invariance and Comparison Principles for Parabolic Stochastic Partial Differential Equations -- Preface -- Chapter 14: White Noise -- 14.1 Some heuristics -- 14.2 LCA groups -- 14.3 White noise on G -- 14.4 Space-time white noise -- 14.5 The Walsh stochastic integral -- 14.5.1 Simple random fields -- 14.5.2 Elementary random fields -- 14.5.3 Walsh-integrable random fields -- 14.6 Moment inequalities -- 14.7 Examples of Walsh-integrable random fields -- 14.7.1 Integral kernels -- 14.7.2 Stochastic convolutions -- 14.7.3 Relation to Itô integrals -- Chapter 15: Lévy Processes -- 15.1 Introduction -- 15.1.1 Lévy processes on R -- 15.1.2 Lévy processes on T -- 15.1.3 Lévy processes on Z -- 15.1.4 Lévy processes on Z/nZ -- 15.2 The semigroup -- 15.3 The Kolmogorov-Fokker-Planck equation -- 15.3.1 Lévy processes on R -- Chapter 16: SPDEs -- 16.1 A heat equation -- 16.2 A parabolic SPDE -- 16.2.1 Lévy processes on R -- 16.2.2 Lévy processes on a denumerable LCA group.
16.2.3 Proof of Theorem 16.2.2 -- 16.3 Examples -- 16.3.1 The trivial group -- 16.3.2 The cyclic group on two elements -- 16.3.3 The integer group -- 16.3.4 The additive reals -- 16.3.5 Higher dimensions -- Chapter 17: An Invariance Principle for Parabolic SPDEs -- 17.1 A central limit theorem -- 17.2 A local central limit theorem -- 17.3 Particle systems -- Chapter 18: Comparison Theorems -- 18.1 Positivity -- 18.2 The Cox-Fleischmann-Greven inequality -- 18.3 Slepian's inequality -- Chapter 19: A Dash of Color -- 19.1 Reproducing kernel Hilbert spaces -- 19.2 Colored noise -- 19.2.1 Example: white noise -- 19.2.2 Example: Hilbert-Schmidt covariance -- 19.2.3 Example: spatially-homogeneous covariance -- 19.2.4 Example: tensor-product covariance -- 19.3 Linear SPDEs with colored-noise forcing -- Bibliography -- Index. |
| Record Nr. | UNINA-9910254090303321 |
Khoshnevisan Davar
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| Cham : , : Springer International Publishing : , : Imprint : Birkhäuser, , 2016 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
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Stochastic Integration by Parts and Functional Itô Calculus / / by Vlad Bally, Lucia Caramellino, Rama Cont ; edited by Frederic Utzet, Josep Vives
| Stochastic Integration by Parts and Functional Itô Calculus / / by Vlad Bally, Lucia Caramellino, Rama Cont ; edited by Frederic Utzet, Josep Vives |
| Autore | Bally Vlad |
| Edizione | [1st ed. 2016.] |
| Pubbl/distr/stampa | Cham : , : Springer International Publishing : , : Imprint : Birkhäuser, , 2016 |
| Descrizione fisica | 1 online resource (IX, 207 p. 1 illus. in color.) |
| Disciplina | 510 |
| Collana | Advanced Courses in Mathematics - CRM Barcelona |
| Soggetto topico |
Probabilities
Differential equations Differential equations, Partial Probability Theory and Stochastic Processes Ordinary Differential Equations Partial Differential Equations |
| ISBN | 3-319-27128-8 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Intro -- Foreword -- Contents -- Part I Integration by Parts Formulas, Malliavin Calculus, and Regularity of Probability Laws -- Preface -- Problem 1 -- Problem 2 -- Problem 3 -- Problem 4 -- Conclusion -- Chapter 1 Integration by parts formulas and the Riesz transform -- 1.1 Sobolev spaces associated to probability measures -- 1.2 The Riesz transform -- 1.3 A first absolute continuity criterion: Malliavin-Thalmaier representation formula -- 1.4 Estimate of the Riesz transform -- 1.5 Regularity of the density -- 1.6 Estimate of the tails of the density -- 1.7 Local integration by parts formulas and local densities -- 1.8 Random variables -- Chapter 2 Construction of integration by parts formulas -- 2.1 Construction of integration by parts formulas -- 2.1.1 Derivative operators -- 2.1.2 Duality and integration by parts formulas -- 2.1.3 Estimation of the weights -- Iterated derivative operators, Sobolev norms -- Estimate of |γ(F)|l -- Bounds for the weights Hqβ (F,G) -- 2.1.4 Norms and weights -- 2.2 Short introduction to Malliavin calculus -- 2.2.1 Differential operators -- Step 1: Finite-dimensional di erential calculus in dimension n -- Step 2: Finite-dimensional di erential calculus in arbitrary dimension -- Step 3: Infinite-dimensional calculus -- 2.2.2 Computation rules and integration by parts formulas -- 2.3 Representation and estimates for the density -- 2.4 Comparisons between density functions -- 2.4.1 Localized representation formulas for the density -- 2.4.2 The distance between density functions -- 2.5 Convergence in total variation for a sequence of Wiener functionals -- Chapter 3 Regularity of probability laws by using an interpolation method -- 3.1 Notations -- 3.2 Criterion for the regularity of a probability law -- 3.3 Random variables and integration by parts -- 3.4 Examples -- 3.4.1 Path dependent SDE's.
3.4.2 Diffusion processes -- 3.4.3 Stochastic heat equation -- 3.5 Appendix A: Hermite expansions and density estimates -- 3.6 Appendix B: Interpolation spaces -- 3.7 Appendix C: Superkernels -- Bibliography -- Part II Functional Itô Calculus and Functional Kolmogorov Equations -- Preface -- Chapter 4 Overview -- 4.1 Functional Itô Calculus -- 4.2 Martingale representation formulas -- 4.3 Functional Kolmogorov equations and path dependent PDEs -- 4.4 Outline -- Notations -- Chapter 5 Pathwise calculus for non-anticipative functionals -- 5.1 Non-anticipative functionals -- 5.2 Horizontal and vertical derivatives -- 5.2.1 Horizontal derivative -- 5.2.2 Vertical derivative -- 5.2.3 Regular functionals -- 5.3 Pathwise integration and functional change of variable formula -- 5.3.1 Quadratic variation of a path along a sequence of partitions -- 5.3.2 Functional change of variable formula -- 5.3.3 Pathwise integration for paths of finite quadratic variation -- 5.4 Functionals defined on continuous paths -- 5.5 Application to functionals of stochastic processes -- Chapter 6 The functional Itô formula -- 6.1 Semimartingales and quadratic variation -- 6.2 The functional Itô formula -- 6.3 Functionals with dependence on quadratic variation -- Chapter 7 Weak functional calculus for square-integrable processes -- 7.1 Vertical derivative of an adapted process -- 7.2 Martingale representation formula -- 7.3 Weak derivative for square integrable functionals -- 7.4 Relation with the Malliavin derivative -- 7.5 Extension to semimartingales -- 7.6 Changing the reference martingale -- 7.7 Forward-Backward SDEs -- Chapter 8 Functional Kolmogorov equations -- 8.1 Functional Kolmogorov equations and harmonic functionals -- 8.1.1 Stochastic differential equations with path dependent coefficients -- 8.1.2 Local martingales and harmonic functionals. 8.1.3 Sub-solutions and super-solutions -- 8.1.4 Comparison principle and uniqueness -- 8.1.5 Feynman-Kac formula for path dependent functionals -- 8.2 FBSDEs and semilinear functional PDEs -- 8.3 Non-Markovian stochastic control and path dependent HJB equations -- 8.4 Weak solutions -- Comments and references -- Bibliography. |
| Record Nr. | UNINA-9910254067503321 |
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Bally Vlad
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| Cham : , : Springer International Publishing : , : Imprint : Birkhäuser, , 2016 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||