Diffusions, Markov processes and martingales: 1 Foundations
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| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge University Press
2006
|
| Ausgabe: | 2. ed., 5. print. |
| Schriftenreihe: | Cambridge mathematical library
|
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XX, 386 S. |
| ISBN: | 0521775949 |
Internformat
MARC
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| 008 | 080829s2006 xx |||| 00||| eng d | ||
| 020 | |a 0521775949 |9 0-521-77594-9 | ||
| 035 | |a (OCoLC)634892392 | ||
| 035 | |a (DE-599)BVBBV035029311 | ||
| 040 | |a DE-604 |b ger |e rakwb | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-739 | ||
| 084 | |a SK 820 |0 (DE-625)143258: |2 rvk | ||
| 100 | 1 | |a Rogers, Leonard C. G. |0 (DE-588)13011359X |4 aut | |
| 245 | 1 | 0 | |a Diffusions, Markov processes and martingales |n 1 |p Foundations |c L. C. G. Rogers and David Williams |
| 250 | |a 2. ed., 5. print. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge University Press |c 2006 | |
| 300 | |a XX, 386 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Cambridge mathematical library | |
| 700 | 1 | |a Williams, David |4 aut | |
| 773 | 0 | 8 | |w (DE-604)BV013208855 |g 1 |
| 856 | 4 | 2 | |m Digitalisierung UB Passau |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016698314&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-016698314 | |
Datensatz im Suchindex
| _version_ | 1816148588990824448 |
|---|---|
| adam_text |
Contents
Some Frequently Used Notation
xix
CHAPTER I. BROWNIAN MOTION
1.
INTRODUCTION
1
1.
What is Brownian motion, and why study it?
1
2.
Brownian motion as a martingale
2
3.
Brownian motion as a Gaussian process
3
4.
Brownian motion as a Markov process
5
5.
Brownian motion as a diffusion (and martingale)
7
2.
BASICS ABOUT BROWNIAN MOTION
10
6.
Existence and uniqueness of Brownian motion
10
7.
Skorokhod embedding
13
8.
Donsker's
Invariance
Principle
16
9.
Exponential martingales and first-passage distributions
18
10.
Some sample-path properties
19
11.
Quadratic variation
21
12.
The strong Markov property
21
13.
Reflection
25
14.
Reflecting Brownian motion and local time
27
15.
Kolmogorov's test
31
16.
Brownian exponential martingales and the Law of the
Iterated Logarithm
31
3.
BROWNIAN MOTION IN HIGHER DIMENSIONS
36
17.
Some martingales for Brownian motion
36
18.
Recurrence and transience in higher dimensions
38
19.
Some applications of Brownian motion to complex analysis
39
20.
Windings of planar Brownian motion
43
21.
Multiple points, cone points, cut points
45
xii CONTENTS
22.
Potential theory of Brownian motion in
R1* {d
^
3) 46
23.
Brownian motion and physical diffusion
51
4.
GAUSSIAN PROCESSES AND LEVY PROCESSES
55
Gaussian processes
24.
Existence results for Gaussian processes
55
25.
Continuity results
59
26. Isotropie
random flows
66
27.
Dynkin's Isomorphism Theorem
71
Levy processes
28.
Levy processes
73
29.
Fluctuation theory and
Wiener-Hopf
factorisation
80
30.
Local time of Levy processes
82
CHAPTER II. SOME CLASSICAL THEORY
1.
BASIC MEASURE THEORY
85
Measurability and measure
1.
Measurable spaces;
σ
-algebras;
π
-systems;
d-systems
85
2.
Measurable functions
88
. 3.
Monotone-Class Theorems
90
4.
Measures; the uniqueness lemma; almost everywhere;
α£.{μ,Σ)
91
5.
Carathéodory's
Extension Theorem
93
6.
Inner and outer
μ
-measures;
completion
94
Integration
7.
Definition of the integral
f
ƒ
άμ
95
8.
Convergence theorems
96
9.
The
Radon-Nikodým
Theorem; absolute continuity;
л
«
μ
notation; equivalent measures
98
10.
Inequalities;
<£'
and
U
spaces (p
> 1) 99
Product structures
11.
Product
σ
-algebras
101
12.
Product measure; Fubini's Theorem
102
13.
Exercises
104
2.
BASIC PROBABILITY THEORY
108
Probability and expectation
14.
Probability triple; almost surely (a.s.); a.s.(P), a.s.(P, 3F)
108
CONTENTS
ХІІІ
15. lim
sup
£„; First
Borei-Cantelli
Lemma 109
16.
Law of random variable; distribution function; joint law
110
17.
Expectation; E(X; F)
110
18.
Inequalities: Markov, Jensen,
Schwarz,
Tchebychev
111
19.
Modes of convergence of random variables
113
Uniform integrability and if1 convergence
20.
Uniform integrabilky
114
21.
Ух
convergence
115
Independence
22.
Independence of
σ
-algebras
and of random variables
116
23.
Existence of families of independent variables
118
24.
Exercises
119
3.
STOCHASTIC PROCESSES
119
The
Danieli-
Kolmogorov Theorem
25.
(ET,<fT);
σ
-algebras
on function space; cylinders and
σ
-cylinders
119
26.
Infinite products of probability triples
121
27.
Stochastic process; sample function; law
121
28.
Canonical process
122
29.
Finite-dimensional distributions; sufficiency; compatibility
123
30.
The Daniell-Kolmogorov (DK) Theorem: 'compact
metrizable' case
124
31.
The Daniell-Kolmogorov (DK) Theorem: general case
126
32.
Gaussian processes; pre-Brownian motion
127
33.
Pre-Poisson set functions
128
Beyond the DK Theorem
34.
Limitations of the DK Theorem
128
35.
The role of outer measures
129
36.
Modifications; indistinguishability
130
37.
Direct construction of
Poisson
measures and subordinators,
and of local time from the zero set;
Azéma's
martingale
131
38.
Exercises
136
4.
DISCRETE-PARAMETER MARTINGALE THEORY
137
Conditional expectation
30.
Fundamental theorem and definition
137
40.
Notation; agreement with elementary usage
138
41.
Properties of conditional expectation: a list
139
42.
The role of versions; regular conditional probabilities and pdfs
140
xiv CONTENTS
43.
A counterexample
141
44.
A uniform-integrability property of conditional expectations
142
(Discrete-parameter) martingales and
supermartingales
45.
Filtration; filtered space; adapted process; natural filtration
143
46.
Martingale;
supermartingale;
submartingale
144
47.
Previsible
process; gambling strategy; a fundamental principle
144
48.
Doob's Upcrossing Lemma
145
49.
Doob's
Supermartingale-Convergence
Theorem
146
50.
¿C1 convergence and the UI property
147
51.
The
Lévy-Doob
Downward Theorem
148
52.
Doob's
Submartingale
and S£p Inequalities
150
53.
Martingales in if2; orthogonality of increments
152
54.
Doob decomposition
153
55.
The <M> and [M] processes
154
Stopping times, optional stopping and optional sampling
56.
Stopping time
155
57.
Optional-stopping theorems
156
58.
The
pre- T
σ
-algebra
&τ
158
59.
Optional sampling
159
60.
Exercises
161
5.
CONTINUOUS-PARAMETER
SUPERMARTINGALES
163
Régularisation: R-supermartingales
61.
Orientation
163
62.
Some real-variable results
163
63.
Filtrations;
supermartingales; R-processes,
R-supermartingales
166
64.
Some important examples
167
65.
Doob's Regularity Theorem: Part
1 169
66.
Partial augmentation
171
67.
Usual conditions; R-filtered space; usual augmentation;
R-regularisation
172
68.
A necessary pause for thought
174
69.
Convergence theorems for R-supermartingales
175
70.
Inequalities and
š£v
convergence for R-submartingales
177
71.
Martingale proof of Wiener's Theorem; canonical
Brownian motion
178
72.
Brownian motion relative to a filtered space
180
Stopping times
73.
Stopping time T;
pre-
Τ σ
-algebra
'S?,
progressive process
181
74.
First-entrance
(début)
times; hitting times; first-approach times:
the easy cases
183
CONTENTS
XV
75.
Why 'completion' in the usual conditions has to be introduced
184
76.
Début
and Section Theorems
186
77.
Optional Sampling for R-supermartingales under the
usual conditions
188
78.
Two important results for Markov-process theory
191
79.
Exercises
192
6.
PROBABILITY MEASURES ON LUSIN SPACES
200
'Weak convergence'
80.
C(J) and Pr(J)wherr
J
is compact Hausdorff
202
81.
C(J) and Pt(J) when
J
is compact metrizable
203
82.
Polish and Lusin spaces
205
83.
The Cb(S) topology of Pr(S) when
S
is a Lusin space;
Prohorov's Theorem
207
84.
Some useful convergence results
211
85.
Tightness in
Рг(И0
when
W
is the path-space W:= C([0, oo); R)
213
86.
The Skorokhod representation of Cb(S) convergence on Pr(S)
215
87.
Weak convergence versus convergence of finite-dimensional
distributions
216
Regular conditional probabilities
88.
Some preliminaries
217
89.
The main existence theorem
218
90.
Canonical Brownian Motion
CBMÍR.");
Markov property of
P* laws
220
91.
Exercises
222
CHAPTER III. MARKOV PROCESSES
1.
TRANSITION FUNCTIONS AND RESOLVENTS
227
1.
What is a (continuous-time) Markov process?
227
2.
The finite-state-space Markov chain
228
3.
Transition functions and their resolvents
231
4.
Contraction semigroups on Banach spaces
234
5.
The Hille-Yosida Theorem
237
2.
FELLER-DYNKIN PROCESSES
240
6.
Feller-Dynkin (FD) semigroups
240
7.
The existence theorem: canonical FD processes
243
8.
Strong Markov property: preliminary version
247
9.
Strong Markov property: full version; Blumenthal's
0-1
Law
249
xvi
CONTENTS
10.
Some fundamental martingales; Dynkin's formula
252
11.
Quasi-left-continuity
255
12.
Characteristic operator
256
13.
Feller-Dynkin diffusions
258
14.
Characterisation of continuous real Levy processes
261
15.
Consolidation
262
3.
ADDITIVE FUNCTIONALS
263
16.
PCHAFs; /-excessive functions; Brownian local time
263
17.
Proof of the
Volkonskii-Šur-Meyer
Theorem
267
18.
Killing
269
19.
The Feynmann-Kac formula
272
20.
A Ciesielski-Taylor Theorem
275
21.
Time-substitution
277
22.
Reflecting Brownian motion
278
23.
The Feller-McKean chain
281
24.
Elastic Brownian motion; the arcsine law
282
4.
APPROACH TO RAY PROCESSES:
THE MARTIN BOUNDARY
284
25.
Ray processes and Markov chains
284
26.
Important example: birth process
286
27.
Excessive functions, the Martin kernel and Choquet theory
288
28.
The Martin compactification
292
29.
The Martin representation; Doob-Hunt explanation
295
30.
R. S. Martin's boundary
297
31.
Doob-Hunt theory for Brownian motion
298
32.
Ray processes and right processes
302
5.
RAY PROCESSES
303
33.
Orientation
303
34.
Ray resolvents
304
35.
The Ray-Knight compactification
306
Ray's Theorem: analytical part
36.
From semigroup to resolvent
309
37.
Branch-points
313
38.
Choquet representation of
1
-excessive probability measures
315
Ray's Theorem: probabilistic part
39.
The Ray process associated with a given entrance law
316
40.
Strong Markov property of Ray processes
318
41.
The role of branch-points
319
CONTENTS xvii
6.
APPLICATIONS
321
Martin
boundary theory in retrospect
42.
From discrete to continuous time
321
43.
Proof of the Doob-Hunt Convergence Theorem
323
44.
The Choquet representation of
П
-excessive functions
325
45.
Doob
Л
-transforms
327
Time reversal and related topics
46.
Nagasawa's formula for chains
328
47.
Strong Markov property under time reversal
330
48.
Equilibrium charge
331
49.
BM(R) and
BES
(3):
splitting times
332
A first look at Markov-chain theory
50.
Chains as Ray processes
334
51.
Significance of
q¡
337
52.
Taboo probabilities; first-entrance decomposition
337
53.
The Q-matrix; DK conditions
339
54.
Local-character condition for
Q
340
55.
Totally instantaneous Q-matrices
342
56.
Last exits
343
57.
Excursions from
b
345
58.
Kingman's solution of the 'Markov characterization problem
347
59.
Symmetrisable chains
348
60.
An open problem
349
References for Volumes
1
and
2 351
Index to Volumes
1
and
2 375 |
| adam_txt |
Contents
Some Frequently Used Notation
xix
CHAPTER I. BROWNIAN MOTION
1.
INTRODUCTION
1
1.
What is Brownian motion, and why study it?
1
2.
Brownian motion as a martingale
2
3.
Brownian motion as a Gaussian process
3
4.
Brownian motion as a Markov process
5
5.
Brownian motion as a diffusion (and martingale)
7
2.
BASICS ABOUT BROWNIAN MOTION
10
6.
Existence and uniqueness of Brownian motion
10
7.
Skorokhod embedding
13
8.
Donsker's
Invariance
Principle
16
9.
Exponential martingales and first-passage distributions
18
10.
Some sample-path properties
19
11.
Quadratic variation
21
12.
The strong Markov property
21
13.
Reflection
25
14.
Reflecting Brownian motion and local time
27
15.
Kolmogorov's test
31
16.
Brownian exponential martingales and the Law of the
Iterated Logarithm
31
3.
BROWNIAN MOTION IN HIGHER DIMENSIONS
36
17.
Some martingales for Brownian motion
36
18.
Recurrence and transience in higher dimensions
38
19.
Some applications of Brownian motion to complex analysis
39
20.
Windings of planar Brownian motion
43
21.
Multiple points, cone points, cut points
45
xii CONTENTS
22.
Potential theory of Brownian motion in
R1* {d
^
3) 46
23.
Brownian motion and physical diffusion
51
4.
GAUSSIAN PROCESSES AND LEVY PROCESSES
55
Gaussian processes
24.
Existence results for Gaussian processes
55
25.
Continuity results
59
26. Isotropie
random flows
66
27.
Dynkin's Isomorphism Theorem
71
Levy processes
28.
Levy processes
73
29.
Fluctuation theory and
Wiener-Hopf
factorisation
80
30.
Local time of Levy processes
82
CHAPTER II. SOME CLASSICAL THEORY
1.
BASIC MEASURE THEORY
85
Measurability and measure
1.
Measurable spaces;
σ
-algebras;
π
-systems;
d-systems
85
2.
Measurable functions
88
. 3.
Monotone-Class Theorems
90
4.
Measures; the uniqueness lemma; almost everywhere;
α£.{μ,Σ)
91
5.
Carathéodory's
Extension Theorem
93
6.
Inner and outer
μ
-measures;
completion
94
Integration
7.
Definition of the integral
f
ƒ
άμ
95
8.
Convergence theorems
96
9.
The
Radon-Nikodým
Theorem; absolute continuity;
л
«
μ
notation; equivalent measures
98
10.
Inequalities;
<£'
and
U
spaces (p
> 1) 99
Product structures
11.
Product
σ
-algebras
101
12.
Product measure; Fubini's Theorem
102
13.
Exercises
104
2.
BASIC PROBABILITY THEORY
108
Probability and expectation
14.
Probability triple; almost surely (a.s.); a.s.(P), a.s.(P, 3F)
108
CONTENTS
ХІІІ
15. lim
sup
£„; First
Borei-Cantelli
Lemma 109
16.
Law of random variable; distribution function; joint law
110
17.
Expectation; E(X; F)
110
18.
Inequalities: Markov, Jensen,
Schwarz,
Tchebychev
111
19.
Modes of convergence of random variables
113
Uniform integrability and if1 convergence
20.
Uniform integrabilky
114
21.
Ух
convergence
115
Independence
22.
Independence of
σ
-algebras
and of random variables
116
23.
Existence of families of independent variables
118
24.
Exercises
119
3.
STOCHASTIC PROCESSES
119
The
Danieli-
Kolmogorov Theorem
25.
(ET,<fT);
σ
-algebras
on function space; cylinders and
σ
-cylinders
119
26.
Infinite products of probability triples
121
27.
Stochastic process; sample function; law
121
28.
Canonical process
122
29.
Finite-dimensional distributions; sufficiency; compatibility
123
30.
The Daniell-Kolmogorov (DK) Theorem: 'compact
metrizable' case
124
31.
The Daniell-Kolmogorov (DK) Theorem: general case
126
32.
Gaussian processes; pre-Brownian motion
127
33.
Pre-Poisson set functions
128
Beyond the DK Theorem
34.
Limitations of the DK Theorem
128
35.
The role of outer measures
129
36.
Modifications; indistinguishability
130
37.
Direct construction of
Poisson
measures and subordinators,
and of local time from the zero set;
Azéma's
martingale
131
38.
Exercises
136
4.
DISCRETE-PARAMETER MARTINGALE THEORY
137
Conditional expectation
30.
Fundamental theorem and definition
137
40.
Notation; agreement with elementary usage
138
41.
Properties of conditional expectation: a list
139
42.
The role of versions; regular conditional probabilities and pdfs
140
xiv CONTENTS
43.
A counterexample
141
44.
A uniform-integrability property of conditional expectations
142
(Discrete-parameter) martingales and
supermartingales
45.
Filtration; filtered space; adapted process; natural filtration
143
46.
Martingale;
supermartingale;
submartingale
144
47.
Previsible
process; gambling strategy; a fundamental principle
144
48.
Doob's Upcrossing Lemma
145
49.
Doob's
Supermartingale-Convergence
Theorem
146
50.
¿C1 convergence and the UI property
147
51.
The
Lévy-Doob
Downward Theorem
148
52.
Doob's
Submartingale
and S£p Inequalities
150
53.
Martingales in if2; orthogonality of increments
152
54.
Doob decomposition
153
55.
The <M> and [M] processes
154
Stopping times, optional stopping and optional sampling
56.
Stopping time
155
57.
Optional-stopping theorems
156
58.
The
pre- T
σ
-algebra
&τ
158
59.
Optional sampling
159
60.
Exercises
161
5.
CONTINUOUS-PARAMETER
SUPERMARTINGALES
163
Régularisation: R-supermartingales
61.
Orientation
163
62.
Some real-variable results
163
63.
Filtrations;
supermartingales; R-processes,
R-supermartingales
166
64.
Some important examples
167
65.
Doob's Regularity Theorem: Part
1 169
66.
Partial augmentation
171
67.
Usual conditions; R-filtered space; usual augmentation;
R-regularisation
172
68.
A necessary pause for thought
174
69.
Convergence theorems for R-supermartingales
175
70.
Inequalities and
š£v
convergence for R-submartingales
177
71.
Martingale proof of Wiener's Theorem; canonical
Brownian motion
178
72.
Brownian motion relative to a filtered space
180
Stopping times
73.
Stopping time T;
pre-
Τ σ
-algebra
'S?,
progressive process
181
74.
First-entrance
(début)
times; hitting times; first-approach times:
the easy cases
183
CONTENTS
XV
75.
Why 'completion' in the usual conditions has to be introduced
184
76.
Début
and Section Theorems
186
77.
Optional Sampling for R-supermartingales under the
usual conditions
188
78.
Two important results for Markov-process theory
191
79.
Exercises
192
6.
PROBABILITY MEASURES ON LUSIN SPACES
200
'Weak convergence'
80.
C(J) and Pr(J)wherr
J
is compact Hausdorff
202
81.
C(J) and Pt(J) when
J
is compact metrizable
203
82.
Polish and Lusin spaces
205
83.
The Cb(S) topology of Pr(S) when
S
is a Lusin space;
Prohorov's Theorem
207
84.
Some useful convergence results
211
85.
Tightness in
Рг(И0
when
W
is the path-space W:= C([0, oo); R)
213
86.
The Skorokhod representation of Cb(S) convergence on Pr(S)
215
87.
Weak convergence versus convergence of finite-dimensional
distributions
216
Regular conditional probabilities
88.
Some preliminaries
217
89.
The main existence theorem
218
90.
Canonical Brownian Motion
CBMÍR.");
Markov property of
P* laws
220
91.
Exercises
222
CHAPTER III. MARKOV PROCESSES
1.
TRANSITION FUNCTIONS AND RESOLVENTS
227
1.
What is a (continuous-time) Markov process?
227
2.
The finite-state-space Markov chain
228
3.
Transition functions and their resolvents
231
4.
Contraction semigroups on Banach spaces
234
5.
The Hille-Yosida Theorem
237
2.
FELLER-DYNKIN PROCESSES
240
6.
Feller-Dynkin (FD) semigroups
240
7.
The existence theorem: canonical FD processes
243
8.
Strong Markov property: preliminary version
247
9.
Strong Markov property: full version; Blumenthal's
0-1
Law
249
xvi
CONTENTS
10.
Some fundamental martingales; Dynkin's formula
252
11.
Quasi-left-continuity
255
12.
Characteristic operator
256
13.
Feller-Dynkin diffusions
258
14.
Characterisation of continuous real Levy processes
261
15.
Consolidation
262
3.
ADDITIVE FUNCTIONALS
263
16.
PCHAFs; /-excessive functions; Brownian local time
263
17.
Proof of the
Volkonskii-Šur-Meyer
Theorem
267
18.
Killing
269
19.
The Feynmann-Kac formula
272
20.
A Ciesielski-Taylor Theorem
275
21.
Time-substitution
277
22.
Reflecting Brownian motion
278
23.
The Feller-McKean chain
281
24.
Elastic Brownian motion; the arcsine law
282
4.
APPROACH TO RAY PROCESSES:
THE MARTIN BOUNDARY
284
25.
Ray processes and Markov chains
284
26.
Important example: birth process
286
27.
Excessive functions, the Martin kernel and Choquet theory
288
28.
The Martin compactification
292
29.
The Martin representation; Doob-Hunt explanation
295
30.
R. S. Martin's boundary
297
31.
Doob-Hunt theory for Brownian motion
298
32.
Ray processes and right processes
302
5.
RAY PROCESSES
303
33.
Orientation
303
34.
Ray resolvents
304
35.
The Ray-Knight compactification
306
Ray's Theorem: analytical part
36.
From semigroup to resolvent
309
37.
Branch-points
313
38.
Choquet representation of
1
-excessive probability measures
315
Ray's Theorem: probabilistic part
39.
The Ray process associated with a given entrance law
316
40.
Strong Markov property of Ray processes
318
41.
The role of branch-points
319
CONTENTS xvii
6.
APPLICATIONS
321
Martin
boundary theory in retrospect
42.
From discrete to continuous time
321
43.
Proof of the Doob-Hunt Convergence Theorem
323
44.
The Choquet representation of
П
-excessive functions
325
45.
Doob
Л
-transforms
327
Time reversal and related topics
46.
Nagasawa's formula for chains
328
47.
Strong Markov property under time reversal
330
48.
Equilibrium charge
331
49.
BM(R) and
BES
(3):
splitting times
332
A first look at Markov-chain theory
50.
Chains as Ray processes
334
51.
Significance of
q¡
337
52.
Taboo probabilities; first-entrance decomposition
337
53.
The Q-matrix; DK conditions
339
54.
Local-character condition for
Q
340
55.
Totally instantaneous Q-matrices
342
56.
Last exits
343
57.
Excursions from
b
345
58.
Kingman's solution of the 'Markov characterization problem
347
59.
Symmetrisable chains
348
60.
An open problem
349
References for Volumes
1
and
2 351
Index to Volumes
1
and
2 375 |
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| author | Rogers, Leonard C. G. Williams, David |
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| illustrated | Not Illustrated |
| index_date | 2024-07-02T21:49:14Z |
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| spelling | Rogers, Leonard C. G. (DE-588)13011359X aut Diffusions, Markov processes and martingales 1 Foundations L. C. G. Rogers and David Williams 2. ed., 5. print. Cambridge [u.a.] Cambridge University Press 2006 XX, 386 S. txt rdacontent n rdamedia nc rdacarrier Cambridge mathematical library Williams, David aut (DE-604)BV013208855 1 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016698314&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Rogers, Leonard C. G. Williams, David Diffusions, Markov processes and martingales |
| title | Diffusions, Markov processes and martingales |
| title_auth | Diffusions, Markov processes and martingales |
| title_exact_search | Diffusions, Markov processes and martingales |
| title_exact_search_txtP | Diffusions, Markov processes and martingales |
| title_full | Diffusions, Markov processes and martingales 1 Foundations L. C. G. Rogers and David Williams |
| title_fullStr | Diffusions, Markov processes and martingales 1 Foundations L. C. G. Rogers and David Williams |
| title_full_unstemmed | Diffusions, Markov processes and martingales 1 Foundations L. C. G. Rogers and David Williams |
| title_short | Diffusions, Markov processes and martingales |
| title_sort | diffusions markov processes and martingales foundations |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016698314&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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