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Markov processes from K. Itô's perspective / / Daniel W. Stroock



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Autore: Stroock Daniel W. Visualizza persona
Titolo: Markov processes from K. Itô's perspective / / Daniel W. Stroock Visualizza cluster
Pubblicazione: Princeton, New Jersey ; ; Oxfordshire, England : , : Princeton University Press, , 2003
©2003
Descrizione fisica: 1 online resource (289 p.)
Disciplina: 519.2/33
Soggetto topico: Markov processes
Stochastic difference equations
Soggetto non controllato: Abelian group
Addition
Analytic function
Approximation
Bernhard Riemann
Bounded variation
Brownian motion
Central limit theorem
Change of variables
Coefficient
Complete metric space
Compound Poisson process
Continuous function (set theory)
Continuous function
Convergence of measures
Convex function
Coordinate system
Corollary
David Hilbert
Decomposition theorem
Degeneracy (mathematics)
Derivative
Diffeomorphism
Differentiable function
Differentiable manifold
Differential equation
Differential geometry
Dimension
Directional derivative
Doob–Meyer decomposition theorem
Duality principle
Elliptic operator
Equation
Euclidean space
Existential quantification
Fourier transform
Function space
Functional analysis
Fundamental solution
Fundamental theorem of calculus
Homeomorphism
Hölder's inequality
Initial condition
Integral curve
Integral equation
Integration by parts
Invariant measure
Itô calculus
Itô's lemma
Joint probability distribution
Lebesgue measure
Linear interpolation
Lipschitz continuity
Local martingale
Logarithm
Markov chain
Markov process
Markov property
Martingale (probability theory)
Normal distribution
Ordinary differential equation
Ornstein–Uhlenbeck process
Polynomial
Principal part
Probability measure
Probability space
Probability theory
Pseudo-differential operator
Radon–Nikodym theorem
Representation theorem
Riemann integral
Riemann sum
Riemann–Stieltjes integral
Scientific notation
Semimartingale
Sign (mathematics)
Special case
Spectral sequence
Spectral theory
State space
State-space representation
Step function
Stochastic calculus
Stochastic
Stratonovich integral
Submanifold
Support (mathematics)
Tangent space
Tangent vector
Taylor's theorem
Theorem
Theory
Topological space
Topology
Translational symmetry
Uniform convergence
Variable (mathematics)
Vector field
Weak convergence (Hilbert space)
Weak topology
Classificazione: SI 830
Note generali: Description based upon print version of record.
Nota di bibliografia: Includes bibliographical references and index.
Nota di contenuto: Frontmatter -- Contents -- Preface -- Chapter 1. Finite State Space, a Trial Run -- Chapter 2. Moving to Euclidean Space, the Real Thing -- Chapter 3. Itô's Approach in the Euclidean Setting -- Chapter 4. Further Considerations -- Chapter 5. Itô's Theory of Stochastic Integration -- Chapter 6. Applications of Stochastic Integration to Brownian Motion -- Chapter 7. The Kunita-Watanabe Extension -- Chapter 8. Stratonovich's Theory -- Notation -- References -- Index
Sommario/riassunto: 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.
Titolo autorizzato: Markov processes from K. Itô's perspective  Visualizza cluster
ISBN: 0-691-11542-7
1-4008-3557-7
Formato: Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione: Inglese
Record Nr.: 9910791958803321
Lo trovi qui: Univ. Federico II
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Serie: Annals of mathematics studies ; ; Number 155.