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Stochastic analysis and diffusion processes /
Stochastic Analysis and Diffusion Processes presents a simple, mathematical introduction to Stochastic Calculus and its applications. The book builds the basic theory and offers a careful account of important research directions in Stochastic Analysis. The breadth and power of Stochastic Analysis, a...
| Clasificación: | Libro Electrónico |
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| Autor principal: | |
| Otros Autores: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Oxford :
Oxford University Press,
2014.
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| Colección: | Oxford graduate texts in mathematics ;
24. |
| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Cover ; Preface; Contents; 1 Introduction to Stochastic Processes; 1.1 The Kolmogorov Consistency Theorem; 1.2 The Language of Stochastic Processes; 1.3 Sigma Fields, Measurability, and Stopping Times; Exercises; 2 Brownian Motion; 2.1 Definition and Construction of Brownian Motion; 2.2 Essential Features of a Brownian Motion; 2.3 The Reflection Principle; Exercises; 3 Elements of Martingale Theory; 3.1 Definition and Examples of Martingales; 3.2 Wiener Martingales and the Markov Property; 3.3 Essential Results on Martingales; 3.4 The Doob-Meyer Decomposition.
- 3.5 The Meyer Process for L2-martingales3.6 Local Martingales; Exercises; 4 Analytical Tools for Brownian Motion; 4.1 Introduction; 4.2 The Brownian Semigroup; 4.3 Resolvents and Generators; 4.4 Pregenerators and Martingales; Exercises; 5 Stochastic Integration; 5.1 The Itô Integral; 5.2 Properties of the Integral; 5.3 Vector-valued Processes; 5.4 The Itô Formula; 5.5 An Extension of the Itô Formula; 5.6 Applications of the Itô Formula; 5.7 The Girsanov Theorem; Exercises; 6 Stochastic Differential Equations; 6.1 Introduction; 6.2 Existence and Uniqueness of Solutions.
- 6.3 Linear Stochastic Differential Equations6.4 Weak Solutions; 6.5 Markov Property; 6.6 Generators and Diffusion Processes; Exercises; 7 The Martingale Problem; 7.1 Introduction; 7.2 Existence of Solutions; 7.3 Analytical Tools; 7.4 Uniqueness of Solutions; 7.5 Markov Property of Solutions; 7.6 Further Results on Uniqueness; 8 Probability Theory and Partial Differential Equations; 8.1 The Dirichlet Problem; 8.2 Boundary Regularity; 8.3 Kolmogorov Equations: The Heuristics; 8.4 Feynman-Kac Formula; 8.5 An Application to Finance Theory; 8.6 Kolmogorov Equations; Exercises; 9 Gaussian Solutions.
- 9.1 Introduction9.2 Hilbert-Schmidt Operators; 9.3 The Gohberg-Krein Factorization; 9.4 Nonanticipative Representations; 9.5 Gaussian Solutions of Stochastic Equations; Exercises; 10 Jump Markov Processes; 10.1 Definitions and Basic Results; 10.2 Stochastic Calculus for Processes with Jumps; 10.3 Jump Markov Processes; 10.4 Diffusion Approximation; Exercises; 11 Invariant Measures and Ergodicity; 11.1 Introduction; 11.2 Ergodicity for One-dimensional Diffusions; 11.3 Invariant Measures for d-dimensional Diffusions; 11.4 Existence and Uniqueness of Invariant Measures; 11.5 Ergodic Measures.
- Exercises12 Large Deviations Principle for Diffusions; 12.1 Definitions and Basic Results; 12.2 Large Deviations and Laplace-Varadhan Principle; 12.3 A Variational Representation Theorem; 12.4 Sufficient Conditions for LDP; Exercises; Notes on Chapters; References; Index.