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

• Autore: Stroock Daniel W.
• 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
• ISBN: 0-691-11542-7; 1-4008-3557-7
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910809577703321
• Lo trovi qui: Univ. Federico II
• Serie: Annals of mathematics studies ; ; Number 155.