Bernt Oksendal: Stochastic differential Equations
Este libro es ideal para ser usado como libro de texto en un curso avanzado, pero también es apropiado para
analistas (en particular, aquellos que trabajan con ecuaciones diferenciales o en sistemas dinámicos
deterministicos) que deseen dominar rápidamente el tema de ecuaciones diferenciales estocásticas.
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Dificultad matemática: *****
Capítulos:
1. Introduction
1.1 Stochastic Analogs of Classical Differential
Equations
1.2 Filtering Problems
1.3 Stochastic Approach to Deterministic Boundary
Value
1.4 Optimal Stopping
1.5 Stochastic Control
1.6 Mathematical Finance
2. Some Mathematical Preliminaries
2.1 Probability Spaces, Random Variables and
Stochastic Processes
2.2 An Important Example: Brownian Motion
3. Ito Integrals
3.1 Construction of the Ito Integral
3.2 Some Properties of the IW Integral
3.3 Extensions of the Ito Integral
4. The Ito Formula and the Martingale Representation orem
4.1 The 1-dimensional Ito Formula
4.2 The Multi-dimensional Ito Formula
4.3 The Martingale Representation Theorem
5. Stochastic Differential Equations
5.1 Examples and Some Solution Methods
5.2 An Existence and Uniqueness Result
5.3 Weak and Strong Solutions
6. The Filtering Problem
6.1 Introduction
6.2 The I-Dimensional Linear Filtering Problem
6.3 The Multidimensional Linear Filtering Problem
7. Diffusions: Basic Properties
7.1 The Markov Property
7.2 The Strong Markov Property
7.3 The Generator of an Ito Diffusion
7.4 The Dynkin Formula
7.5 The Characteristic Operator
8. Other Topics in Diffusion Theory
8.1 Kolmogorov's Backward Equation. The Resolvent
8.2 The Feynman-Kac Formula. Killing
8.3 The Martingale Problem
8.4 When is an Ito Process a Diffusion?
8.5 Random Time Change
8.6 The Girsanov Theorem
9. Applications to Boundary Value Problems
9.1 The Combined Dirichlet-Poisson Problem.
Uniqueness
9.2 The Dirichlet Problem. Regular Points
9.3 The Poisson Problem
10. Application to Optimal Stopping
10.1 The Time-Homogeneous Case ...
10.2 The Time-Inhomogeneous Case
10.3 Optimal Stopping Problems Involving an Integral
10.4 Connection with Variational Inequalities
11. Application to Stochastic Control
11.1 Statement of the Problem
11.2 The Hamilton-Jacobi-Bellman Equation
11.3 Stochastic Control Problems with Terminal
Conditions
12. Application to Mathematical Finance
12.1 Market, Portfolio and Arbitrage
12.2 Attainability and Completeness
12.3 Option Pricing
Appendix A: Normal Random Variables
Appendix B: Conditional Expectation
Appendix C: Uniform Integrability and Martingale
Convergence
Appendix D: An Approximation Result