Markov Processes from K. Itô's Perspective
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A01=Daniel W. Stroock
Addition
Analytic function
Author_Daniel W. Stroock
Bernhard Riemann
Bounded variation
Brownian motion
Category=PBK
Category=PBM
Category=PBWL
Central limit theorem
Change of variables
Coefficient
Compound Poisson process
Continuous function
Continuous function (set theory)
Coordinate system
David Hilbert
Degeneracy (mathematics)
Derivative
Differentiable function
Differential equation
Differential geometry
Doob-Meyer decomposition theorem
Elliptic operator
eq_isMigrated=1
eq_isMigrated=2
eq_nobargain
Equation
Existential quantification
Fourier transform
Function space
Functional analysis
Fundamental solution
Fundamental theorem of calculus
Homeomorphism
Integral curve
Integral equation
Integration by parts
Ito calculus
Ito's lemma
Joint probability distribution
Lebesgue measure
Lipschitz continuity
Markov chain
Markov process
Markov property
Martingale (probability theory)
Ordinary differential equation
Ornstein-Uhlenbeck process
Polynomial
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)
Spectral theory
State-space representation
Step function
Stochastic
Stochastic calculus
Stratonovich integral
Submanifold
Tangent space
Taylor's theorem
Theorem
Theory
Topology
Translational symmetry
Vector field
Weak convergence (Hilbert space)
Product details
- ISBN 9780691115436
- Weight: 397g
- Dimensions: 152 x 235mm
- Publication Date: 26 May 2003
- Publisher: Princeton University Press
- Publication City/Country: US
- Product Form: Paperback
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Kiyosi Ito'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 Ito'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, Ito 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 Ito'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 Ito's stochastic integral calculus. In the second half, the author provides a systematic development of Ito's theory of stochastic integration: first for Brownian motion and then for continuous martingales. The final chapter presents Stratonovich's variation on Ito'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.
Daniel W. Stroock is a Simons Professor of Mathematics at the Massachusetts Institute of Technology and the author of several books, including "A Concise Introduction to the Theory of Integration and Probability Theory, an Analytic View".
