Stochastic stability of differential equations:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Weitere Verfasser: | , |
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2012
|
| Ausgabe: | Completely rev. and enl. 2. ed. |
| Schriftenreihe: | Stochastic modelling and applied probability
66 |
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | Aus dem Russ. übers. |
| Beschreibung: | XVII, 339 S. 24 cm |
| ISBN: | 9783642232794 3642232795 9783642232800 |
Internformat
MARC
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| 245 | 1 | 0 | |a Stochastic stability of differential equations |c Rafail Khasminskii. With contributions by G. N. Milstein and M. B. Nevelson |
| 250 | |a Completely rev. and enl. 2. ed. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2012 | |
| 300 | |a XVII, 339 S. |c 24 cm | ||
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Datensatz im Suchindex
| _version_ | 1805145450594959360 |
|---|---|
| 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 |
| any_adam_object | 1 |
| author | Chasʹminskij, Rafail Z. |
| author2 | Milʹstejn, Grigorij N. 1937- Nevel'son, Michail B. |
| author2_role | edt edt |
| author2_variant | g n m gn gnm m b n mb mbn |
| author_GND | (DE-588)1018362622 (DE-588)12144371X |
| author_facet | Chasʹminskij, Rafail Z. Milʹstejn, Grigorij N. 1937- Nevel'son, Michail B. |
| author_role | aut |
| author_sort | Chasʹminskij, Rafail Z. |
| author_variant | r z c rz rzc |
| building | Verbundindex |
| bvnumber | BV039664745 |
| classification_rvk | SK 820 |
| classification_tum | MAT 606f |
| ctrlnum | (OCoLC)759585460 (DE-599)DNB1013760905 |
| dewey-full | 515.392 519.22 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis 519 - Probabilities and applied mathematics |
| dewey-raw | 515.392 519.22 |
| dewey-search | 515.392 519.22 |
| dewey-sort | 3515.392 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Completely rev. and enl. 2. ed. |
| format | Book |
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| id | DE-604.BV039664745 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-21T00:14:13Z |
| institution | BVB |
| isbn | 9783642232794 3642232795 9783642232800 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-024514050 |
| oclc_num | 759585460 |
| open_access_boolean | |
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| owner_facet | DE-11 DE-83 DE-824 DE-19 DE-BY-UBM DE-91G DE-BY-TUM DE-20 |
| physical | XVII, 339 S. 24 cm |
| publishDate | 2012 |
| publishDateSearch | 2012 |
| publishDateSort | 2012 |
| publisher | Springer |
| record_format | marc |
| series | Stochastic modelling and applied probability |
| series2 | Stochastic modelling and applied probability |
| 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 |