Holdings: Stochastic stability of differential equations

Stochastic stability of differential equations:

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Bibliographic Details
Main Author:	Chasʹminskij, Rafail Z. (Author)
Other Authors:	Milʹstejn, Grigorij N. 1937- (Editor), Nevel'son, Michail B. (Editor)
Format:	Book
Language:	English
Published:	Berlin [u.a.] Springer 2012
Edition:	Completely rev. and enl. 2. ed.
Series:	Stochastic modelling and applied probability 66
Subjects:	Qualitative Theorie Stabilität Stochastische Differentialgleichung Ljapunov-Exponent
Online Access:	Inhaltstext Inhaltsverzeichnis
Item Description:	Aus dem Russ. übers.
Physical Description:	XVII, 339 S. 24 cm
ISBN:	9783642232794 3642232795 9783642232800

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Record in the Search Index

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adam_text	IMAGE 1 CONTENTS 1 BOUNDEDNESS IN PROBABILITY AND STABILITY OF STOCHASTIC PROCESSES DENNED BY DIFFERENTIAL EQUATIONS 1 1.1 BRIEF REVIEW OF PREREQUISITES FROM PROBABILITY THEORY 1 1.2 DISSIPATIVE SYSTEMS OF DIFFERENTIAL EQUATIONS 4 1.3 STOCHASTIC PROCESSES AS SOLUTIONS OF DIFFERENTIAL EQUATIONS . . . 9 1.4 BOUNDEDNESS IN PROBABILITY OF STOCHASTIC PROCESSES DEFINED BY SYSTEMS OF DIFFERENTIAL EQUATIONS 13 1.5 STABILITY 22 1.6 STABILITY OF RANDOMLY PERTURBED DETERMINISTIC SYSTEMS 26 1.7 ESTIMATION OF A CERTAIN FUNCTIONAL OF A GAUSSIAN PROCESS 31 1.8 LINEAR SYSTEMS 36 2 STATIONARY AND PERIODIC SOLUTIONS OF DIFFERENTIAL EQUATIONS 43 2.1 STATIONARY AND PERIODIC STOCHASTIC PROCESSES. CONVERGENCE OF STOCHASTIC PROCESSES 43 2.2 EXISTENCE CONDITIONS FOR STATIONARY AND PERIODIC SOLUTIONS . . . 46 2.3 SPECIAL EXISTENCE CONDITIONS FOR STATIONARY AND PERIODIC SOLUTIONS 51 2.4 CONDITIONS FOR CONVERGENCE TO A PERIODIC SOLUTION 55 3 MARKOV PROCESSES AND STOCHASTIC DIFFERENTIAL EQUATIONS 59 3.1 DEFINITION OF MARKOV PROCESSES 59 3.2 STATIONARY AND PERIODIC MARKOV PROCESSES 63 3.3 STOCHASTIC DIFFERENTIAL EQUATIONS (SDE) 67 3.4 CONDITIONS FOR REGULARITY OF THE SOLUTION 74 3.5 STATIONARY AND PERIODIC SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS 79 3.6 STOCHASTIC EQUATIONS AND PARTIAL DIFFERENTIAL EQUATIONS 83 3.7 CONDITIONS FOR RECURRENCE AND FINITENESS OF MEAN RECURRENCE TIME 89 3.8 FURTHER CONDITIONS FOR RECURRENCE AND FINITENESS OF MEAN RECURRENCE TIME 93 BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1013760905 DIGITALISIERT DURCH IMAGE 2 CONTENTS ERGODIC PROPERTIES OF SOLUTIONS OF STOCHASTIC EQUATIONS 99 4.1 KOLMOGOROV CLASSIFICATION OF MARKOV CHAINS WITH COUNTABLY MANY STATES 99 4.2 RECURRENCE AND TRANSIENCE 101 4.3 POSITIVE AND NULL RECURRENT PROCESSES 105 4.4 EXISTENCE OF A STATIONARY DISTRIBUTION 106 4.5 STRONG LAW OF LARGE NUMBERS 109 4.6 SOME AUXILIARY RESULTS 112 4.7 EXISTENCE OF THE LIMIT OF THE TRANSITION PROBABILITY FUNCTION . . 117 4.8 SOME GENERALIZATIONS 119 4.9 STABILIZATION OF THE SOLUTION OF THE CAUCHY PROBLEM FOR A PARABOLIC EQUATION 122 4.10 LIMIT RELATIONS FOR NULL RECURRENT PROCESSES 127 4.11 LIMIT RELATIONS FOR NULL RECURRENT PROCESSES (CONTINUED) 131 4.12 ARCSINE LAW AND ONE GENERALIZATION 136 STABILITY OF STOCHASTIC DIFFERENTIAL EQUATIONS 145 5.1 STATEMENT OF THE PROBLEM 145 5.2 SOME AUXILIARY RESULTS 148 5.3 STABILITY IN PROBABILITY 152 5.4 ASYMPTOTIC STABILITY IN PROBABILITY AND INSTABILITY 155 5.5 EXAMPLES 159 5.6 DIFFERENTIABILITY OF SOLUTIONS OF STOCHASTIC EQUATIONS WITH RESPECT TO THE INITIAL CONDITIONS 165 5.7 EXPONENTIAL P-STABILITY AND Q -INSTABILITY 171 5.8 ALMOST SURE EXPONENTIAL STABILITY 175 SYSTEMS OF LINEAR STOCHASTIC EQUATIONS 177 6.1 ONE-DIMENSIONAL SYSTEMS 177 6.2 EQUATIONS FOR MOMENTS 182 6.3 EXPONENTIAL P-STABILITY AND ^-INSTABILITY 184 6.4 EXPONENTIAL P-STABILITY AND ^-INSTABILITY (CONTINUED) 188 6.5 UNIFORM STABILITY IN THE LARGE 192 6.6 STABILITY OF PRODUCTS OF INDEPENDENT MATRICES 196 6.7 ASYMPTOTIC STABILITY OF LINEAR SYSTEMS WITH CONSTANT COEFFICIENTS 201 6.8 SYSTEMS WITH CONSTANT COEFFICIENTS (CONTINUED) 206 6.9 TWO EXAMPLES 211 6.10 N-TH ORDER EQUATIONS 216 6.11 STOCHASTIC STABILITY IN THE STRONG AND WEAK SENSES 223 SOME SPECIAL PROBLEMS IN THE THEORY OF STABILITY OF SDE'S 227 7.1 STABILITY IN THE FIRST APPROXIMATION 227 7.2 INSTABILITY IN THE FIRST APPROXIMATION 229 7.3 TWO EXAMPLES 231 7.4 STABILITY UNDER DAMPED RANDOM PERTURBATIONS 234 7.5 APPLICATION TO STOCHASTIC APPROXIMATION 237 IMAGE 3 CONTENTS XV 7.6 STOCHASTIC APPROXIMATIONS WHEN THE REGRESSION EQUATION HAS SEVERAL ROOTS 239 7.7 SOME GENERALIZATIONS 245 7.7.1 STABILITY AND EXCESSIVE FUNCTIONS 245 7.7.2 STABILITY OF THE INVARIANT SET 247 7.7.3 EQUATIONS WHOSE COEFFICIENTS ARE MARKOV PROCESSES . . . 247 7.7.4 STABILITY UNDER PERSISTENT PERTURBATION BY WHITE NOISE . . 249 7.7.5 BOUNDEDNESS IN PROBABILITY OF THE OUTPUT PROCESS OF A NONLINEAR STOCHASTIC SYSTEM 251 8 STABILIZATION OF CONTROLLED STOCHASTIC SYSTEMS (THIS CHAPTER WAS WRITTEN JOINTLY WITH M.B. NEVELSON) 253 8.1 PRELIMINARY REMARKS 253 8.2 BELLMAN'S PRINCIPLE 254 8.3 LINEAR SYSTEMS 258 8.4 METHOD OF SUCCESSIVE APPROXIMATIONS 260 APPENDIX A APPENDIX TO THE FIRST ENGLISH EDITION 265 A. 1 MOMENT STABILITY AND ALMOST SURE STABILITY FOR LINEAR SYSTEMS OF EQUATIONS WHOSE COEFFICIENTS ARE MARKOV PROCESSES 265 A.2 ALMOST SURE STABILITY OF THE PATHS OF ONE-DIMENSIONAL DIFFUSION PROCESSES 269 A.3 REDUCTION PRINCIPLE 275 A.4 SOME FURTHER RESULTS 279 APPENDIX B APPENDIX TO THE SECOND EDITION. MOMENT LYAPUNOV EXPONENTS AND STABILITY INDEX (WRITTEN JOINTLY WITH G.N. MILSTEIN) 281 B.I PRELIMINARIES 281 B.2 BASIC THEOREMS 285 B.2.1 NONDEGENERACY CONDITIONS 285 B.2.2 SEMIGROUPS OF POSITIVE COMPACT OPERATORS AND MOMENT LYAPUNOV EXPONENTS 286 B.2.3 GENERATOR OF THE PROCESS A 294 B.2.4 GENERATOR OF SEMIGROUP T,(P)F(K) 296 B.2.5 VARIOUS REPRESENTATIONS OF SEMIGROUP T T (P)F(\) 299 B.3 STABILITY INDEX 303 B.3.1 STABILITY INDEX FOR LINEAR STOCHASTIC DIFFERENTIAL EQUATIONS 303 B.3.2 STABILITY INDEX FOR NONLINEAR SDES 305 B.4 MOMENT LYAPUNOV EXPONENT AND STABILITY INDEX FOR SYSTEM WITH SMALL NOISE 309 B.4.1 INTRODUCTION AND STATEMENT OF PROBLEM 309 B.4.2 METHOD OF ASYMPTOTIC EXPANSION 312 B.4.3 STABILITY INDEX 316 B.4.4 APPLICATIONS 319 REFERENCES 323 INDEX 335
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author	Chasʹminskij, Rafail Z.
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spelling	Chasʹminskij, Rafail Z. Verfasser (DE-588)1018362622 aut Stochastic stability of differential equations Rafail Khasminskii. With contributions by G. N. Milstein and M. B. Nevelson Completely rev. and enl. 2. ed. Berlin [u.a.] Springer 2012 XVII, 339 S. 24 cm txt rdacontent n rdamedia nc rdacarrier Stochastic modelling and applied probability 66 Aus dem Russ. übers. Qualitative Theorie (DE-588)4571165-3 gnd rswk-swf Stabilität (DE-588)4056693-6 gnd rswk-swf Stochastische Differentialgleichung (DE-588)4057621-8 gnd rswk-swf Ljapunov-Exponent (DE-588)4123668-3 gnd rswk-swf Stochastische Differentialgleichung (DE-588)4057621-8 s Qualitative Theorie (DE-588)4571165-3 s Ljapunov-Exponent (DE-588)4123668-3 s DE-604 Stabilität (DE-588)4056693-6 s 1\p DE-604 Milʹstejn, Grigorij N. 1937- (DE-588)12144371X edt Nevel'son, Michail B. edt Erscheint auch als Online-Ausgabe Stochastic Stability of Differential Equations Stochastic modelling and applied probability 66 (DE-604)BV019623501 66 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3857324&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024514050&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk
spellingShingle	Chasʹminskij, Rafail Z. Stochastic stability of differential equations Stochastic modelling and applied probability Qualitative Theorie (DE-588)4571165-3 gnd Stabilität (DE-588)4056693-6 gnd Stochastische Differentialgleichung (DE-588)4057621-8 gnd Ljapunov-Exponent (DE-588)4123668-3 gnd
subject_GND	(DE-588)4571165-3 (DE-588)4056693-6 (DE-588)4057621-8 (DE-588)4123668-3
title	Stochastic stability of differential equations
title_auth	Stochastic stability of differential equations
title_exact_search	Stochastic stability of differential equations
title_full	Stochastic stability of differential equations Rafail Khasminskii. With contributions by G. N. Milstein and M. B. Nevelson
title_fullStr	Stochastic stability of differential equations Rafail Khasminskii. With contributions by G. N. Milstein and M. B. Nevelson
title_full_unstemmed	Stochastic stability of differential equations Rafail Khasminskii. With contributions by G. N. Milstein and M. B. Nevelson
title_short	Stochastic stability of differential equations
title_sort	stochastic stability of differential equations
topic	Qualitative Theorie (DE-588)4571165-3 gnd Stabilität (DE-588)4056693-6 gnd Stochastische Differentialgleichung (DE-588)4057621-8 gnd Ljapunov-Exponent (DE-588)4123668-3 gnd
topic_facet	Qualitative Theorie Stabilität Stochastische Differentialgleichung Ljapunov-Exponent
url	http://deposit.dnb.de/cgi-bin/dokserv?id=3857324&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024514050&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV019623501
work_keys_str_mv	AT chasʹminskijrafailz stochasticstabilityofdifferentialequations AT milʹstejngrigorijn stochasticstabilityofdifferentialequations AT nevelsonmichailb stochasticstabilityofdifferentialequations

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