Markov Processes from K. Ito's Perspective (AM-155).
Title:
Markov Processes from K. Ito's Perspective (AM-155).
Author:
Stroock, Daniel W.
ISBN:
9781400835577
Personal Author:
Physical Description:
1 online resource (289 pages)
Series:
Annals of Mathematics Studies
Contents:
Cover -- Title -- Copyright -- Dedication -- Contents -- Preface -- Chapter 1 Finite State Space, a Trial Run -- 1.1 An Extrinsic Perspective -- 1.1.1. The Structure of Θn -- 1.1.2. Back to M1(Zn) -- 1.2 A More Intrinsic Approach -- 1.2.1. The Semigroup Structure on M1(Zn) -- 1.2.2. Infinitely Divisible Flows -- 1.2.3. An Intrinsic Description of Tδx (M1(Zn)) -- 1.2.4. An Intrinsic Approach to (1.1.6) -- 1.2.5. Exercises -- 1.3 Vector Fields and Integral Curves on M1(Zn) -- 1.3.1. Affine and Translation Invariant Vector Fields -- 1.3.2. Existence of an Integral Curve -- 1.3.3. Uniqueness for Affine Vector Fields -- 1.3.4. The Markov Property and Kolmogorov's Equations -- 1.3.5. Exercises -- 1.4 Pathspace Realization -- 1.4.1. Kolmogorov's Approach -- 1.4.2. Lévy Processes on Zn -- 1.4.3. Exercises -- 1.5 Itô's Idea -- 1.5.1. Itô's Construction -- 1.5.2. Exercises -- 1.6 Another Approach -- 1.6.1. Itô's Approximation Scheme -- 1.6.2. Exercises -- Chapter 2 Moving to Euclidean Space, the Real Thing -- 2.1 Tangent Vectors to M1(R^n) -- 2.1.1. Differentiable Curves on M1(R^n) -- 2.1.2. Infinitely Divisible Flows on M1(R^n) -- 2.1.3. The Tangent Space at δx -- 2.1.4. The Tangent Space at General μ ∈ M1(R^n) -- 2.1.5. Exercises -- 2.2 Vector Fields and Integral Curves on M1(R^n) -- 2.2.1. Existence of Integral Curves -- 2.2.2. Uniqueness for Affine Vector Fields -- 2.2.3. The Markov Property and Kolmogorov's Equations -- 2.2.4. Exercises -- 2.3 Pathspace Realization, Preliminary Version -- 2.3.1. Kolmogorov's Construction -- 2.3.2. Path Regularity -- 2.3.3. Exercises -- 2.4 The Structure of Levy Processes on R^n -- 2.4.1. Construction -- 2.4.2. Structure -- 2.4.3. Exercises -- Chapter 3 Itô's Approach in the Euclidean Setting -- 3.1 Itô's Basic Construction -- 3.1.1. Transforming Lévy Processes -- 3.1.2. Hypotheses and Goals.
3.1.3. Important Preliminary Observations -- 3.1.4. The Proof of Convergence -- 3.1.5. Verifying the Martingale Property in (G2) -- 3.1.6. Exercises -- 3.2 When Does Itô's Theory Work? -- 3.2.1. The Diffusion Coefficients -- 3.2.2. The Lévy Measure -- 3.2.3. Exercises -- 3.3 Some Examples to Keep in Mind -- 3.3.1. The Ornstein-Uhlenbeck Process -- 3.3.2. Bachelier's Model -- 3.3.3. A Geometric Example -- 3.3.4. Exercises -- Chapter 4 Further Considerations -- 4.1 Continuity, Measurability, and the Markov Property -- 4.1.1. Continuity and Measurability -- 4.1.2. The Markov Property -- 4.1.3. Exercises -- 4.2 Differentiability -- 4.2.1. First Derivatives -- 4.2.2. Second Derivatives and Uniqueness -- Chapter 5 Itô's Theory of Stochastic Integration -- 5.1 Brownian Stochastic Integrals -- 5.1.1. A Review of the Paley-Wiener Integral -- 5.1.2. Itô's Extension -- 5.1.3. Stopping Stochastic Integrals and a Further Extension -- 5.1.4. Exercises -- 5.2 Itô's Integral Applied to Itô's Construction Method -- 5.2.1. Existence and Uniqueness -- 5.2.2. Subordination -- 5.2.3. Exercises -- 5.3 Itô's Formula -- 5.3.1. Exercises -- Chapter 6 Applications of Stochastic Integration to Brownian Motion -- 6.1 Tanaka's Formula for Local Time -- 6.1.1. Tanaka's Construction -- 6.1.2. Some Properties of Local Time -- 6.1.3. Exercises -- 6.2 An Extension of the Cameron-Martin Formula -- 6.2.1. Introduction of a Random Drift -- 6.2.2. An Application to Pinned Brownian Motion -- 6.2.3. Exercises -- 6.3 Homogeneous Chaos -- 6.3.1. Multiple Stochastic Integrals -- 6.3.2. The Spaces of Homogeneous Chaos -- 6.3.3. Exercises -- Chapter 7 The Kunita-Watanabe Extension -- 7.1 Doob-Meyer for Continuous Martingales -- 7.1.1. Uniqueness -- 7.1.2. Existence -- 7.1.3. Exercises -- 7.2 Kunita-Watanabe Stochastic Integration -- 7.2.1. The Hilbert Structure of Mloc(P -- R).
7.2.2. The Kunita-Watanabe Stochastic Integral -- 7.2.3. General Itô's Formula -- 7.2.4. Exercises -- 7.3 Representations of Continuous Martingales -- 7.3.1. Representation via Random Time Change -- 7.3.2. Representation via Stochastic Integration -- 7.3.3. Skorohod's Representation Theorem -- 7.3.4. Exercises -- Chapter 8 Stratonovich's Theory -- 8.1 Semimartingales and Stratonovich Integrals -- 8.1.1. Semimartingales -- 8.1.2. Stratonovich's Integral -- 8.1.3. Ito's Formula and Stratonovich Integration -- 8.1.4. Exercises -- 8.2 Stratonovich Stochastic Differential Equations -- 8.2.1. Commuting Vector Fields -- 8.2.2. General Vector Fields -- 8.2.3. Another Interpretation -- 8.2.4. Exercises -- 8.3 The Support Theorem -- 8.3.1. The Support Theorem, Part I -- 8.3.2. The Support Theorem, Part II -- 8.3.3. The Support Theorem, Part III -- 8.3.4. The Support Theorem, Part IV -- 8.3.5. The Support Theorem, Part V -- 8.3.6. Exercises -- Notation -- References -- Index.
Abstract:
Kiyosi Itô's greatest contribution to probability theory may be his introduction of stochastic differential equations to explain the Kolmogorov-Feller theory of Markov processes. Starting with the geometric ideas that guided him, this book gives an account of Itô's program. The modern theory of Markov processes was initiated by A. N. Kolmogorov. However, Kolmogorov's approach was too analytic to reveal the probabilistic foundations on which it rests. In particular, it hides the central role played by the simplest Markov processes: those with independent, identically distributed increments. To remedy this defect, Itô interpreted Kolmogorov's famous forward equation as an equation that describes the integral curve of a vector field on the space of probability measures. Thus, in order to show how Itô's thinking leads to his theory of stochastic integral equations, Stroock begins with an account of integral curves on the space of probability measures and then arrives at stochastic integral equations when he moves to a pathspace setting. In the first half of the book, everything is done in the context of general independent increment processes and without explicit use of Itô's stochastic integral calculus. In the second half, the author provides a systematic development of Itô's theory of stochastic integration: first for Brownian motion and then for continuous martingales. The final chapter presents Stratonovich's variation on Itô's theme and ends with an application to the characterization of the paths on which a diffusion is supported. The book should be accessible to readers who have mastered the essentials of modern probability theory and should provide such readers with a reasonably thorough introduction to continuous-time, stochastic processes.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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