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Table of Contents
Intro
Preface
A Trilogy on Stochastic Processes
Contents
1 A Review of Martingales, Stopping Times, and the Markov Property
Exercises
2 Semigroup Theory and Markov Processes
Exercises
3 Regularity of Markov Process Sample Paths
Exercises
4 Continuous Parameter Jump Markov Processes
Exercises
5 Processes with Independent Increments
Exercises
6 The Stochastic Integral
Exercises
7 Construction of Diffusions as Solutions of Stochastic Differential Equations
7.1 Construction of One-Dimensional Diffusions
7.2 Extension to Multidimensional Diffusions
7.3 An Extension of the Itô Integral & SDEs with Locally Lipschitz Coefficients
7.4 Strong Markov Property
7.5 An Extension to SDEs with Nonhomogeneous Coefficients
7.6 An Extension to k-Dimensional SDE Governed by r-Dimensional Brownian Motion
Exercises
8 Itô's Lemma
8.1 Asymptotic Properties of One-Dimensional Diffusions: Transience and Recurrence
Exercises
9 Cameron-Martin-Girsanov Theorem
Exercises
10 Support of Nonsingular Diffusions
Exercises
11 Transience and Recurrence of Multidimensional Diffusions
Exercises
12 Criteria for Explosion
Exercises
13 Absorption, Reflection, and Other Transformations of Markov Processes
13.1 Absorption
13.2 General One-Dimensional Diffusions on Half-Line with Absorption at Zero
13.3 Reflecting Diffusions
Exercises
14 The Speed of Convergence to Equilibrium of Discrete Parameter Markov Processes and Diffusions
Exercises
15 Probabilistic Representation of Solutions to Certain PDEs
15.1 Feynman-Kaĉ Formula for Multidimensional Diffusion
15.2 Kolmogorov Forward Equation (The Fokker-Planck Equation)
Exercises
16 Probabilistic Solution of the Classical Dirichlet Problem
Exercises
17 The Functional Central Limit Theorem for Ergodic Markov Processes
17.1 A Functional Central Limit Theorem for Diffusions with Periodic Coefficients
Exercises
18 Asymptotic Stability for Singular Diffusions
Exercises
19 Stochastic Integrals with L2-Martingales
Exercises
20 Local Time for Brownian Motion
Exercises
21 Construction of One-Dimensional Diffusions by Semigroups
Exercises
22 Eigenfunction Expansions of Transition Probabilities for One-Dimensional Diffusions
Exercises
23 Special Topic: The Martingale Problem
Exercises
24 Special Topic: Multiphase Homogenization for Transport in Periodic Media
Exercises
25 Special Topic: Skew Random Walk and Skew Brownian Motion
Exercises
26 Special Topic: Piecewise Deterministic Markov Processes in Population Biology
Exercises
A The Hille-Yosida Theorem and Closed Graph Theorem
References
Related Textbooks and Monographs
Symbol Index
Author Index
Subject Index
Preface
A Trilogy on Stochastic Processes
Contents
1 A Review of Martingales, Stopping Times, and the Markov Property
Exercises
2 Semigroup Theory and Markov Processes
Exercises
3 Regularity of Markov Process Sample Paths
Exercises
4 Continuous Parameter Jump Markov Processes
Exercises
5 Processes with Independent Increments
Exercises
6 The Stochastic Integral
Exercises
7 Construction of Diffusions as Solutions of Stochastic Differential Equations
7.1 Construction of One-Dimensional Diffusions
7.2 Extension to Multidimensional Diffusions
7.3 An Extension of the Itô Integral & SDEs with Locally Lipschitz Coefficients
7.4 Strong Markov Property
7.5 An Extension to SDEs with Nonhomogeneous Coefficients
7.6 An Extension to k-Dimensional SDE Governed by r-Dimensional Brownian Motion
Exercises
8 Itô's Lemma
8.1 Asymptotic Properties of One-Dimensional Diffusions: Transience and Recurrence
Exercises
9 Cameron-Martin-Girsanov Theorem
Exercises
10 Support of Nonsingular Diffusions
Exercises
11 Transience and Recurrence of Multidimensional Diffusions
Exercises
12 Criteria for Explosion
Exercises
13 Absorption, Reflection, and Other Transformations of Markov Processes
13.1 Absorption
13.2 General One-Dimensional Diffusions on Half-Line with Absorption at Zero
13.3 Reflecting Diffusions
Exercises
14 The Speed of Convergence to Equilibrium of Discrete Parameter Markov Processes and Diffusions
Exercises
15 Probabilistic Representation of Solutions to Certain PDEs
15.1 Feynman-Kaĉ Formula for Multidimensional Diffusion
15.2 Kolmogorov Forward Equation (The Fokker-Planck Equation)
Exercises
16 Probabilistic Solution of the Classical Dirichlet Problem
Exercises
17 The Functional Central Limit Theorem for Ergodic Markov Processes
17.1 A Functional Central Limit Theorem for Diffusions with Periodic Coefficients
Exercises
18 Asymptotic Stability for Singular Diffusions
Exercises
19 Stochastic Integrals with L2-Martingales
Exercises
20 Local Time for Brownian Motion
Exercises
21 Construction of One-Dimensional Diffusions by Semigroups
Exercises
22 Eigenfunction Expansions of Transition Probabilities for One-Dimensional Diffusions
Exercises
23 Special Topic: The Martingale Problem
Exercises
24 Special Topic: Multiphase Homogenization for Transport in Periodic Media
Exercises
25 Special Topic: Skew Random Walk and Skew Brownian Motion
Exercises
26 Special Topic: Piecewise Deterministic Markov Processes in Population Biology
Exercises
A The Hille-Yosida Theorem and Closed Graph Theorem
References
Related Textbooks and Monographs
Symbol Index
Author Index
Subject Index