Diffusions, Markov processes and martingales: 2 Itô calculus
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| Format: | Book |
| Language: | English |
| Published: |
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Cambridge University Press
2008
|
| Edition: | 2. ed., 6. print. |
| Series: | Cambridge mathematical library
|
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 480 S. |
| ISBN: | 9780521775939 |
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| 020 | |a 9780521775939 |9 978-0-521-77593-9 | ||
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| 084 | |a SK 520 |0 (DE-625)143244: |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 2 |p Itô calculus |c L. C. G. Rogers and David Williams |
| 250 | |a 2. ed., 6. print. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge University Press |c 2008 | |
| 300 | |a XIII, 480 S. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 0 | |a Cambridge mathematical library | |
| 700 | 1 | |a Williams, David |4 aut | |
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Contents
Some Frequently Used Notation
xiv
CHAPTER IV. INTRODUCTION TO
ITO
CALCULUS
TERMINOLOGY AND CONVENTIONS
R-processes and L-processes
Usual conditions, etc.
Important convention about time
0
1.
SOME MOTIVATING REMARKS
1.
Ito
integrals
. 2
2.
Integration by parts
. 4
3.
Itô's
formula for Brownian motion
. 8
4.
A rough plan of the chapter
. 9
2.
SOME FUNDAMENTAL IDEAS:
PREVISIBLE
PROCESSES,
LOCALIZATION, etc.
Previsible
processes
5.
Basic integrands Z(S,
Г]
. 10
6.
Previsible
processes on
(0,
oo),
<?,
bJ",
Ьїї
. 11
Finite-variation and integrable-variation processes
1.
FV0 and IV0 processes
. 14
8.
Preservation of the martingale property
. 14
Localization
9.
Я(0, Г], Хт
. 15
10.
Localization of integrands, lb^
. 16
11.
Localization of integrators,
^#0,100,
F
V^#01oc etc
. 17
12.
Nil desperandum!
. 18
13.
Extending stochastic integrals by localization
. 20
14.
Local martingales,
^ťioe,
and the Fatou lemma
. 21
Semimartingales as integrators
15.
Semimartingales,^
. 23
16.
Integrators
. 24
Likelihood ratios
17.
Martingale property under change of measure
. 25
CONTENTS
IX
3.
THE ELEMENTARY THEORY OF FINITE-VARIATION
PROCESSES
18.
Itô's
formula for FV functions
. 27
19.
The
Doléans
exponential
8*
(χ.).
29
Applications to Markov chains with finite state-space
20.
Martingale problems
. 30
21.
Probabilistic interpretation of
β
. 33
22.
Likelihood ratios and some key distributions
. 37
4.
STOCHASTIC INTEGRALS: THE L2 THEORY
23.
Orientation
. 42
24.
Stable spaces of M\,
сЈЃ0,
ÅM\.
42
25.
Elementary stochastic integrals relative to
M
in Ji\
. 45
26.
The processes [M] and [M, N]
. 46
27.
Constructing stochastic integrals in L2
. 47
28.
The Kunita-Watanabe inequalities
. 50
5.
STOCHASTIC INTEGRALS WITH RESPECT TO
CONTINUOUS SEMIMARTINGALES
29.
Orientation
. 52
30.
Quadratic variation for continuous local martingales
. 52
31.
Canonical decomposition of a continuous
semimartingale.
. . 57
32.
Itô's
formula for continuous semimartingales
. 58
6.
APPLICATIONS OF
ITÔ'S
FORMULA
33.
Levy's theorem
. 63
34.
Continuous local martingales as time-changes of Brownian
motion
. 64
35.
Bessel processes; skew products; etc
. 69
36.
Brownian martingale representation
. 73
37.
Exponential semimartingales; estimates
. 75
38.
Cameron-Martin-Girsanov change of measure
. 79
39.
First applications: Doob /i-transforms; hitting of spheres; etc.
. 83
40.
Further applications: bridges; excursions; etc
. 86
41.
Explicit Brownian martingale representation
. 89
42.
Burkholder-Davis-Gundy inequalities
. 93
43.
Semimartingale
local time; Tanaka's formula
. 95
44.
Study of joint continuity
. 99
45.
Local time as an occupation density; generalized
Itô-Tanaka
formula
. 102
46.
The Stratonovich calculus
. 106
47.
Riemann-sum approximation to
Ito
and Stratonovich integrals;
simulation
. 108
X
CONTENTS
CHAPTER V. STOCHASTIC DIFFERENTIAL EQUATIONS AND
DIFFUSIONS
1.
INTRODUCTION
1.
What is a diffusion in
WÌ
. 110
2. FD
diffusions recalled
. 112
3.
SDEs as a means of constructing diffusions
. 113
4.
Example: Brownian motion on a surface
. 114
5.
Examples: modelling noise in physical systems
. 114
6.
Example: Skorokhod's equation
. 117
7.
Examples: control problems
. 119
2.
PATHWISE UNIQUENESS, STRONG SDEs, AND FLOWS
8.
Our general SDE;
previsible
path functionals; diffusion SDEs
. 122
9.
Path wise uniqueness; exact SDEs
. 124
10.
Relationship between exact SDEs and strong solutions
. . . . 125
11.
The
Ito
existence and uniqueness result
. 128
12.
Locally Lipschitz SDEs; Lipschitz properties of a1/2
. 132
13.
Flows; the diffeomorj
"
-лі
theorem; time-reversed flows
. 136
14.
Carverhill's noisy North-South flow on a circle
. 141
15.
The martingale optimality principle in control
. 144
3.
WEAK SOLUTIONS, UNIQUENESS IN LAW
16.
Weak solutions of SDEs; Tanaka's SDE
. 149
17.
'Exact equals weak plus pathwise unique'
. 151
18.
Tsirel'son's example
. 155
4.
MARTINGALE PROBLEMS, MARKOV PROPERTY
19.
Definition; orientation
. 158
20.
Equivalence of the martingale-problem and 'weak' formulations
160
21.
Martingale problems and the strong Markov property
. 162
22.
Appraisal and consolidation: where we have reached
. 163
23.
Existence of solutions to the martingale problem
. 166
24.
The Stroock-
Varadhan
uniqueness theorem
. 170
25.
Martingale representation
. 173
Transformation of SDEs
26.
Change of time scale; Girsanov's SDE
. 175
27.
Change of measure
. 177
28.
Change of state-space; scale; Zvonkin's observation; the Doss-
Sussmann method
. 178
29.
Krylov's example
. 181
5.
OVERTURE TO STOCHASTIC DIFFERENTIAL GEOMETRY
30.
Introduction; some key ideas;
Stratonovich-to-Itô
conversion
. 182
31.
Brownian motion on a submanifold of UN
. 186
CONTENTS Xl
32. Parallel
displacement; Riemannian connections.
193
33.
Extrinsic theory of BMh0r(O(Z)); rolling without slipping; martin¬
gales on manifolds; etc
. 198
34.
Intrinsic theory; normal coordinates; structural equations; diffu¬
sions on manifolds; etc.(!)
. 203
35.
Brownian motion on Lie groups
. 224
36.
Dynkin's Brownian motion of ellipses; hyperbolic space interpret¬
ation; etc
. 239
37.
Khasminskii's method for studying stability; random vibrations.
246
38. Hörmander's
theorem; Malliavin calculus; stochastic pullback;
curvature
. 250
6.
ONE-DIMENSIONAL SDEs
39.
A local-time criterion for pathwise uniqueness
. 263
40.
The Yamada-Watanabe pathwise uniqueness theorem
. 265
41.
The Nakao pathwise-uniqueness theorem
. 266
42.
Solution of a variance control problem
. 267
43.
A comparison theorem
. 269
7.
ONE-DIMENSIONAL DIFFUSIONS
44.
Orientation
. 270
45.
Regular diffusions
. 271
46.
The scale function,
s
. 273
47.
The speed measure, m; time substitution
. 276
48.
Example: the Bessel SDE
. 284
49.
Diffusion local time
.: . 289
50.
Analytical aspects
. 291
51.
Classification of boundary points
. 295
52.
Khasminskii's test for explosion
. 297
53.
An ergodic theorem for 1-dimensional diffusions
. 300
54.
Coupling of 1-dimensional diffusions
. 301
CHAPTER VI. THE GENERAL THEORY
1.
ORIENTATION
1.
Preparatory remarks
. 304
2.
Levy processes
. 308
2.
DEBUT AND SECTION THEOREMS
3.
Progressive processes
. 313
4.
Optional processes,
Û
;
optional times
. 315
5.
The Optional' section theorem
. 317
6.
Warning (not to be skipped)
. 318
Xli CONTENTS
3. OPTIONAL
PROJECTIONS
AND FILTERING
7.
Optional projection °X of X
. 319
8.
The innovations approach to filtering
. 322
9.
The Kalman-Bucy filter
. 327
10.
The Bayesian approach to filtering; a change-detection filter
. . 329
11.
Robust filtering
. 331
4.
CHARACTERIZING
PREVISIBLE
TIMES
12.
Previsible
stopping times; PFA theorem
. 332
13.
Totally inaccessible and accessible stopping times
. 334
14.
Some examples
. 336
15.
Meyer's previsibility theorem for Markov processes
. 338
16.
Proof of the PFA theorem
. 340
17.
Theff-algebras J^do-), jT(p),
У(р + )
. 343
18.
Quasi-left-continuous nitrations
. 346
5.
DUAL
PREVISIBLE
PROJECTIONS
19.
The
previsible
section theorem; the
previsible
projection PX
of*
. 347
20.
Doléans'
characterization of FV processes
. 349
21.
Dual
previsible
projections, compensators
. 350
22.
Cumulative risk
. 352
23.
Some Brownian motion examples
. 354
24.
Decomposition of a continuous
semimartingale
. 358
25.
Proof of the basic
(μ,
A) correspondence
. 359
26.
Proof of the
Doléans
'optional' characterization result
. 360
27.
Proof of the
Doléans
'previsible'
characterization result
. . . 361
28.
Levy systems for Markov processes
. 364
6.
THE MEYER DECOMPOSITION THEOREM
29.
Introduction
. 367
30.
The
Doléans
proof of the Meyer decomposition
. 369
31.
Regular class (D)
submartingales;
approximation to compensators
372
32.
The local form of the decomposition theorem
. 374
33.
An L2 bounded local martingale which is not a martingale
. . 375
34.
The <M> process
. 376
35.
Last exits and equilibrium charge
. 377
7.
STOCHASTIC INTEGRATION: THE GENERAL CASE
36.
The quadratic variation process [M]
. 382
37.
Stochastic integrals with respect to local martingales
. 388
38.
Stochastic integrals with respect to semimartingales
. 391
39.
Itô's
formula for semimartingales
. 394
40.
Special semimartingales
. 394
41.
Quasimartingales
. 396
CONTENTS
ХШ
8.
ITÔ EXCURSION
THEORY
42.
Introduction
. 398
43.
Excursion theory for a finite Markov chain
. 400
44.
Taking stock
. 405
45.
Local time
L
at a regular extremal point
а
. 406
46.
Some technical points:
hypothèses droites,
etc
. 410
47.
The
Poisson
point process of excursions
. 413
48.
Markovian character of
n
. 416
49.
Marking the excursions
. 418
50.
Last-exit decomposition; calculation of the excursion law
и
. . 420
51.
The Skorokhod embedding theorem
. 425
52.
Diffusion properties of local time in the space variable; the
Ray-Knight theorem
. 428
53.
Arcsine law for Brownian motion
. 431
54.
Resolvent density of a
1
-dimensional diffusion
. 432
55.
Path decomposition of Brownian motions and of excursions
. . 433
56.
An illustrative calculation
. 438
57.
Feller Brownian motions
. 439
58.
Example: censoring and reweighting of excursion laws
. 442
59.
Excursion theory by stochastic calculus:
McGilľs
lemma
. . . 445
REFERENCES
. 449
INDEX
. 469 |
| adam_txt |
Contents
Some Frequently Used Notation
xiv
CHAPTER IV. INTRODUCTION TO
ITO
CALCULUS
TERMINOLOGY AND CONVENTIONS
R-processes and L-processes
Usual conditions, etc.
Important convention about time
0
1.
SOME MOTIVATING REMARKS
1.
Ito
integrals
. 2
2.
Integration by parts
. 4
3.
Itô's
formula for Brownian motion
. 8
4.
A rough plan of the chapter
. 9
2.
SOME FUNDAMENTAL IDEAS:
PREVISIBLE
PROCESSES,
LOCALIZATION, etc.
Previsible
processes
5.
Basic integrands Z(S,
Г]
. 10
6.
Previsible
processes on
(0,
oo),
<?,
bJ",
Ьїї
. 11
Finite-variation and integrable-variation processes
1.
FV0 and IV0 processes
. 14
8.
Preservation of the martingale property
. 14
Localization
9.
Я(0, Г], Хт
. 15
10.
Localization of integrands, lb^
. 16
11.
Localization of integrators,
^#0,100,
F
V^#01oc etc
. 17
12.
Nil desperandum!
. 18
13.
Extending stochastic integrals by localization
. 20
14.
Local martingales,
^ťioe,
and the Fatou lemma
. 21
Semimartingales as integrators
15.
Semimartingales,^
. 23
16.
Integrators
. 24
Likelihood ratios
17.
Martingale property under change of measure
. 25
CONTENTS
IX
3.
THE ELEMENTARY THEORY OF FINITE-VARIATION
PROCESSES
18.
Itô's
formula for FV functions
. 27
19.
The
Doléans
exponential
8*
(χ.).
29
Applications to Markov chains with finite state-space
20.
Martingale problems
. 30
21.
Probabilistic interpretation of
β
. 33
22.
Likelihood ratios and some key distributions
. 37
4.
STOCHASTIC INTEGRALS: THE L2 THEORY
23.
Orientation
. 42
24.
Stable spaces of M\,
сЈЃ0,
ÅM\.
42
25.
Elementary stochastic integrals relative to
M
in Ji\
. 45
26.
The processes [M] and [M, N]
. 46
27.
Constructing stochastic integrals in L2
. 47
28.
The Kunita-Watanabe inequalities
. 50
5.
STOCHASTIC INTEGRALS WITH RESPECT TO
CONTINUOUS SEMIMARTINGALES
29.
Orientation
. 52
30.
Quadratic variation for continuous local martingales
. 52
31.
Canonical decomposition of a continuous
semimartingale.
. . 57
32.
Itô's
formula for continuous semimartingales
. 58
6.
APPLICATIONS OF
ITÔ'S
FORMULA
33.
Levy's theorem
. 63
34.
Continuous local martingales as time-changes of Brownian
motion
. 64
35.
Bessel processes; skew products; etc
. 69
36.
Brownian martingale representation
. 73
37.
Exponential semimartingales; estimates
. 75
38.
Cameron-Martin-Girsanov change of measure
. 79
39.
First applications: Doob /i-transforms; hitting of spheres; etc.
. 83
40.
Further applications: bridges; excursions; etc
. 86
41.
Explicit Brownian martingale representation
. 89
42.
Burkholder-Davis-Gundy inequalities
. 93
43.
Semimartingale
local time; Tanaka's formula
. 95
44.
Study of joint continuity
. 99
45.
Local time as an occupation density; generalized
Itô-Tanaka
formula
. 102
46.
The Stratonovich calculus
. 106
47.
Riemann-sum approximation to
Ito
and Stratonovich integrals;
simulation
. 108
X
CONTENTS
CHAPTER V. STOCHASTIC DIFFERENTIAL EQUATIONS AND
DIFFUSIONS
1.
INTRODUCTION
1.
What is a diffusion in
WÌ
. 110
2. FD
diffusions recalled
. 112
3.
SDEs as a means of constructing diffusions
. 113
4.
Example: Brownian motion on a surface
. 114
5.
Examples: modelling noise in physical systems
. 114
6.
Example: Skorokhod's equation
. 117
7.
Examples: control problems
. 119
2.
PATHWISE UNIQUENESS, STRONG SDEs, AND FLOWS
8.
Our general SDE;
previsible
path functionals; diffusion SDEs
. 122
9.
Path wise uniqueness; exact SDEs
. 124
10.
Relationship between exact SDEs and strong solutions
. . . . 125
11.
The
Ito
existence and uniqueness result
. 128
12.
Locally Lipschitz SDEs; Lipschitz properties of a1/2
. 132
13.
Flows; the diffeomorj
"
-лі
theorem; time-reversed flows
. 136
14.
Carverhill's noisy North-South flow on a circle
. 141
15.
The martingale optimality principle in control
. 144
3.
WEAK SOLUTIONS, UNIQUENESS IN LAW
16.
Weak solutions of SDEs; Tanaka's SDE
. 149
17.
'Exact equals weak plus pathwise unique'
. 151
18.
Tsirel'son's example
. 155
4.
MARTINGALE PROBLEMS, MARKOV PROPERTY
19.
Definition; orientation
. 158
20.
Equivalence of the martingale-problem and 'weak' formulations
160
21.
Martingale problems and the strong Markov property
. 162
22.
Appraisal and consolidation: where we have reached
. 163
23.
Existence of solutions to the martingale problem
. 166
24.
The Stroock-
Varadhan
uniqueness theorem
. 170
25.
Martingale representation
. 173
Transformation of SDEs
26.
Change of time scale; Girsanov's SDE
. 175
27.
Change of measure
. 177
28.
Change of state-space; scale; Zvonkin's observation; the Doss-
Sussmann method
. 178
29.
Krylov's example
. 181
5.
OVERTURE TO STOCHASTIC DIFFERENTIAL GEOMETRY
30.
Introduction; some key ideas;
Stratonovich-to-Itô
conversion
. 182
31.
Brownian motion on a submanifold of UN
. 186
CONTENTS Xl
32. Parallel
displacement; Riemannian connections.
193
33.
Extrinsic theory of BMh0r(O(Z)); rolling without slipping; martin¬
gales on manifolds; etc
. 198
34.
Intrinsic theory; normal coordinates; structural equations; diffu¬
sions on manifolds; etc.(!)
. 203
35.
Brownian motion on Lie groups
. 224
36.
Dynkin's Brownian motion of ellipses; hyperbolic space interpret¬
ation; etc
. 239
37.
Khasminskii's method for studying stability; random vibrations.
246
38. Hörmander's
theorem; Malliavin calculus; stochastic pullback;
curvature
. 250
6.
ONE-DIMENSIONAL SDEs
39.
A local-time criterion for pathwise uniqueness
. 263
40.
The Yamada-Watanabe pathwise uniqueness theorem
. 265
41.
The Nakao pathwise-uniqueness theorem
. 266
42.
Solution of a variance control problem
. 267
43.
A comparison theorem
. 269
7.
ONE-DIMENSIONAL DIFFUSIONS
44.
Orientation
. 270
45.
Regular diffusions
. 271
46.
The scale function,
s
. 273
47.
The speed measure, m; time substitution
. 276
48.
Example: the Bessel SDE
. 284
49.
Diffusion local time
.: . 289
50.
Analytical aspects
. 291
51.
Classification of boundary points
. 295
52.
Khasminskii's test for explosion
. 297
53.
An ergodic theorem for 1-dimensional diffusions
. 300
54.
Coupling of 1-dimensional diffusions
. 301
CHAPTER VI. THE GENERAL THEORY
1.
ORIENTATION
1.
Preparatory remarks
. 304
2.
Levy processes
. 308
2.
DEBUT AND SECTION THEOREMS
3.
Progressive processes
. 313
4.
Optional processes,
Û
;
optional times
. 315
5.
The Optional' section theorem
. 317
6.
Warning (not to be skipped)
. 318
Xli CONTENTS
3. OPTIONAL
PROJECTIONS
AND FILTERING
7.
Optional projection °X of X
. 319
8.
The innovations approach to filtering
. 322
9.
The Kalman-Bucy filter
. 327
10.
The Bayesian approach to filtering; a change-detection filter
. . 329
11.
Robust filtering
. 331
4.
CHARACTERIZING
PREVISIBLE
TIMES
12.
Previsible
stopping times; PFA theorem
. 332
13.
Totally inaccessible and accessible stopping times
. 334
14.
Some examples
. 336
15.
Meyer's previsibility theorem for Markov processes
. 338
16.
Proof of the PFA theorem
. 340
17.
Theff-algebras J^do-), jT(p),
У(р + )
. 343
18.
Quasi-left-continuous nitrations
. 346
5.
DUAL
PREVISIBLE
PROJECTIONS
19.
The
previsible
section theorem; the
previsible
projection PX
of*
. 347
20.
Doléans'
characterization of FV processes
. 349
21.
Dual
previsible
projections, compensators
. 350
22.
Cumulative risk
. 352
23.
Some Brownian motion examples
. 354
24.
Decomposition of a continuous
semimartingale
. 358
25.
Proof of the basic
(μ,
A) correspondence
. 359
26.
Proof of the
Doléans
'optional' characterization result
. 360
27.
Proof of the
Doléans
'previsible'
characterization result
. . . 361
28.
Levy systems for Markov processes
. 364
6.
THE MEYER DECOMPOSITION THEOREM
29.
Introduction
. 367
30.
The
Doléans
proof of the Meyer decomposition
. 369
31.
Regular class (D)
submartingales;
approximation to compensators
372
32.
The local form of the decomposition theorem
. 374
33.
An L2 bounded local martingale which is not a martingale
. . 375
34.
The <M> process
. 376
35.
Last exits and equilibrium charge
. 377
7.
STOCHASTIC INTEGRATION: THE GENERAL CASE
36.
The quadratic variation process [M]
. 382
37.
Stochastic integrals with respect to local martingales
. 388
38.
Stochastic integrals with respect to semimartingales
. 391
39.
Itô's
formula for semimartingales
. 394
40.
Special semimartingales
. 394
41.
Quasimartingales
. 396
CONTENTS
ХШ
8.
ITÔ EXCURSION
THEORY
42.
Introduction
. 398
43.
Excursion theory for a finite Markov chain
. 400
44.
Taking stock
. 405
45.
Local time
L
at a regular extremal point
а
. 406
46.
Some technical points:
hypothèses droites,
etc
. 410
47.
The
Poisson
point process of excursions
. 413
48.
Markovian character of
n
. 416
49.
Marking the excursions
. 418
50.
Last-exit decomposition; calculation of the excursion law
и
. . 420
51.
The Skorokhod embedding theorem
. 425
52.
Diffusion properties of local time in the space variable; the
Ray-Knight theorem
. 428
53.
Arcsine law for Brownian motion
. 431
54.
Resolvent density of a
1
-dimensional diffusion
. 432
55.
Path decomposition of Brownian motions and of excursions
. . 433
56.
An illustrative calculation
. 438
57.
Feller Brownian motions
. 439
58.
Example: censoring and reweighting of excursion laws
. 442
59.
Excursion theory by stochastic calculus:
McGilľs
lemma
. . . 445
REFERENCES
. 449
INDEX
. 469 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Rogers, Leonard C. G. Williams, David |
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| illustrated | Not Illustrated |
| index_date | 2024-07-02T21:49:14Z |
| indexdate | 2024-11-19T11:04:24Z |
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| isbn | 9780521775939 |
| language | English |
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| spelling | Rogers, Leonard C. G. (DE-588)13011359X aut Diffusions, Markov processes and martingales 2 Itô calculus L. C. G. Rogers and David Williams 2. ed., 6. print. Cambridge [u.a.] Cambridge University Press 2008 XIII, 480 S. txt rdacontent n rdamedia nc rdacarrier Cambridge mathematical library Williams, David aut (DE-604)BV013208855 2 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016698323&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 2 Itô calculus L. C. G. Rogers and David Williams |
| title_fullStr | Diffusions, Markov processes and martingales 2 Itô calculus L. C. G. Rogers and David Williams |
| title_full_unstemmed | Diffusions, Markov processes and martingales 2 Itô calculus L. C. G. Rogers and David Williams |
| title_short | Diffusions, Markov processes and martingales |
| title_sort | diffusions markov processes and martingales ito calculus |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016698323&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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