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Markov processes, Feller semigroups and evolution equations / Jan A. van Casteren

Author
Casteren, J. A. van
Published
Singapore ; Hackensack, NJ : World Scientific, 2011.
Physical Description
xviii, 805 pages ; 24 cm.

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Contents
Machine generated contents note: 1.Introduction: Stochastic differential equations -- 1.1.Weak and strong solutions to stochastic differential equations -- 1.2.Stochastic differential equations in the infinite-dimensional setting -- 1.3.Martingales -- 1.4.Operator-valued Brownian motion and the Heston volatility model -- 1.5.Stopping times and time-homogeneous Markov processes -- Strong Markov Processes -- 2.Strong Markov processes on Polish spaces -- 2.1.Strict topology -- 2.1.1.Theorem of Daniell-Stone -- 2.1.2.Measures on Polish spaces -- 2.1.3.Integral operators on the space of bounded continuous functions -- 2.2.Strong Markov processes and Feller evolutions -- 2.2.1.The operators -- 2.2.2.Generators of Markov processes and maximum principles -- 2.3.Strong Markov processes: Main results -- 2.3.1.Some historical remarks and references -- 2.4.Dini's lemma, Scheffe's theorem, and the monotone class theorem -- 2.4.1.Dini's lemma and Scheffe's theorem -- 2.4.2.Monotone class theorem -- 2.4.3.Some additional information -- 3.Strong Markov processes: Proof of main results -- 3.1.Proof of the main results: Theorems 2.9 through 2.13 -- 3.1.1.Proof of Theorem 2.9 -- 3.1.2.Proof of Theorem 2.10 -- 3.1.3.Proof of Theorem 2.11 -- 3.1.4.Proof of Theorem 2.12 -- 3.1.5.Proof of Theorem 2.13 -- 3.1.6.Some historical remarks -- 3.1.7.Kolmogorov extension theorem -- 4.Space-time operators and miscellaneous topics -- 4.1.Space-time operators -- 4.2.Dissipative operators and maximum principle -- 4.3.Korovkin property -- 4.4.Continuous sample paths -- 4.5.Measurability properties of hitting times -- 4.5.1.Some related remarks -- Backward Stochastic Differential Equations -- 5.Feynman-Kac formulas, backward stochastic differential equations and Markov processes -- 5.1.Introduction -- 5.2.A probabilistic approach: Weak solutions -- 5.2.1.Some more explanation -- 5.3.Existence and uniqueness of solutions to BSDE's -- 5.4.Backward stochastic differential equations and Markov processes -- 5.4.1.Remarks on the Runge-Kutta method and on monotone operators -- 6.Viscosity solutions, backward stochastic differential equations and Markov processes -- 6.1.Comparison theorems -- 6.2.Viscosity solutions -- 6.3.Backward stochastic differential equations in finance -- 6.4.Some related remarks -- 7.The Hamilton-Jacobi-Bellman equation and the stochastic Noether theorem -- 7.1.Introduction -- 7.2.The Hamilton-Jacobi-Bellman equation and its solution -- 7.3.The Hamilton-Jacobi-Bellman equation and viscosity solutions -- 7.4.A stochastic Noether theorem -- 7.4.1.Classical Noether theorem -- 7.4.2.Some problems -- Long Time Behavior -- 8.On non-stationary Markov processes and Dunford projections -- 8.1.Introduction -- 8.2.Kolmogorov operators and weak*-continuous semigroups -- 8.3.Kolmogorov operators and analytic semigroups -- 8.3.1.Ornstein-Uhlenbeck process -- 8.3.2.Some stochastic differential equations -- 8.4.Ergodicity in the non-stationary case -- 8.5.Conclusions -- 8.6.Another characterization of generators of analytic semigroups -- 8.7.A version of the Bismut-Elworthy formula -- 9.Coupling methods and Sobolev type inequalities -- 9.1.Coupling methods -- 9.2.Some ergodic theorems -- 9.3.Spectral gap -- 9.4.Some related stability results -- 9.5.Notes -- 10.Invariant measure -- 10.1.Markov Chains: Invariant measure -- 10.1.1.Some definitions and results -- 10.2.Markov processes and invariant measures -- 10.2.1.Some additional relevant results -- 10.2.2.An attempt to construct an invariant measure -- 10.2.3.Auxiliary results -- 10.2.4.Actual construction of an invariant measure -- 10.3.A proof of Orey's theorem -- 10.4.About invariant (or stationary) measures -- 10.4.1.Possible applications -- 10.4.2.Conclusion.
Subject(s)
ISBN
9789814322188 (hbk.)
9814322180 (hbk.)
Bibliography Note
Includes bibliographical references (pages 759-788) and index.
Source of Acquisition
UP-PAT copy: Purchased with funds from the Paterno Libraries Endowment; 2010.
Endowment Note
Paterno Libraries Endowment
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