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Markov Processes, Feller Semigroups And Evolution Equations : Series on Concrete & Applicable Mathematics - Jan A.Van Casteren

Markov Processes, Feller Semigroups And Evolution Equations

Series on Concrete & Applicable Mathematics


Published: 25th November 2010
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The book provides a systemic treatment of time-dependent strong Markov processes with values in a Polish space. It describes its generators and the link with stochastic differential equations in infinite dimensions. In a unifying way, where the square gradient operator is employed, new results for backward stochastic differential equations and long-time behavior are discussed in depth. The book also establishes a link between propagators or evolution families with the Feller property and time-inhomogeneous Markov processes. This mathematical material finds its applications in several branches of the scientific world, among which are mathematical physics, hedging models in financial mathematics, and population models.

Prefacep. vii
Introductionp. 1
Introduction: Stochastic differential equationsp. 3
Weak and strong solutions to stochastic differential equationsp. 3
Stochastic differential equations in the infinite-dimensional settingp. 45
Martingalesp. 90
Operator-valued Brownian motion and the Heston volatility modelp. 95
Stopping times and time-homogeneous Markov processesp. 104
Strong Markov Processesp. 107
Strong Markov processes on Polish spacesp. 109
Strict topologyp. 109
Theorem of Daniell-Stonep. 110
Measures on Polish spacesp. 116
Integral operators on the space of bounded continuous functionsp. 128
Strong Markov processes and Feller evolutionsp. 138
The operators ¿t, $$$t and $$$tp. 142
Generators of Markov processes and maximum principlesp. 143
Strong Markov processes: Main resultsp. 147
Some historical remarks and referencesp. 158
Dini's lemma, Scheffé's theorem, and the monotone class theoremp. 159
Dini's lemma and Scheffé's theoremp. 159
Monotone class theoremp. 162
Some additional informationp. 164
Strong Markov processes: Proof of main resultsp. 167
Proof of the main results: Theorems 2.9 through 2.13p. 167
Proof of Theorem 2.9p. 167
Proof of Theorem 2.10p. 192
Proof of Theorem 2.11p. 195
Proof of Theorem 2.12p. 199
Proof of Theorem 2.13p. 219
Some historical remarksp. 222
Kolmogorov extension theoremp. 224
Space-time operators and miscellaneous topicsp. 227
Space-time operatorsp. 227
Dissipative operators and maximum principlep. 240
Korovkin propertyp. 260
Continuous sample pathsp. 280
Measurability properties of hitting timesp. 282
Some related remarksp. 299
Backward Stochastic Differential Equationsp. 301
Feynman-Kac formulas, backward stochastic differential equations and Markov processesp. 303
Introductionp. 304
A probabilistic approach: Weak solutionsp. 327
Some more explanationp. 330
Existence and uniqueness of solutions to BSDE'sp. 335
Backward stochastic differential equations and Markov processesp. 371
Remarks on the Runge-Kutta method and on monotone operatorsp. 379
Viscosity solutions, backward stochastic differential equations and Markov processesp. 385
Comparison theoremsp. 386
Viscosity solutionsp. 392
Backward stochastic differential equations in financep. 399
Some related remarksp. 405
The Hamilton-Jacobi-Bellman equation and the stochastic Noether theoremp. 407
Introductionp. 407
The Hamilton-Jacobi-Bellman equation and its solutionp. 411
The Hamilton-Jacobi-Bellman equation and viscosity solutionsp. 420
A stochastic Noether theoremp. 436
Classical Noether theoremp. 446
Some problemsp. 448
Long Time Behaviorp. 451
On non-stationary Markov processes and Dunford projectionsp. 453
Introductionp. 453
Kolmogorov operators and weak*-continuous semigroupsp. 455
Kolmogorov operators and analytic semigroupsp. 460
Ornstein-Uhlenbeck processp. 477
Some stochastic differential equationsp. 503
Ergodicity in the non-stationary casep. 518
Conclusionsp. 537
Another characterization of generators of analytic semigroupsp. 543
A version of the Bismut-Elworthy formulap. 550
Coupling methods and Sobolev type inequalitiesp. 555
Coupling methodsp. 555
Some ergodic theoremsp. 597
Spectral gapp. 602
Some related stability resultsp. 611
Notesp. 644
Invariant measurep. 647
Markov Chains: Invariant measurep. 647
Some definitions and resultsp. 648
Markov processes and invariant measuresp. 660
Some additional relevant resultsp. 665
An attempt to construct an invariant measurep. 671
Auxiliary resultsp. 684
Actual construction of an invariant measurep. 702
A proof of Orey's theoremp. 731
About invariant (or stationary) measuresp. 752
Possible applicationsp. 754
Conclusionp. 754
Bibliographyp. 759
Indexp. 789
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9789814322188
ISBN-10: 9814322180
Series: Series on Concrete & Applicable Mathematics
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 824
Published: 25th November 2010
Publisher: World Scientific Publishing Co Pte Ltd
Country of Publication: SG
Dimensions (cm): 22.86 x 15.88  x 4.57
Weight (kg): 1.28