Numerical integration of stochastic differential equations:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Dordrecht u.a.
Kluwer
1995
|
| Series: | Mathematics and its applications
313 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Aus dem Russ. übers. |
| Physical Description: | VII, 169 S. |
| ISBN: | 079233213X |
Staff View
MARC
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| 100 | 1 | |a Milʹstejn, Grigorij N. |d 1937- |e Verfasser |0 (DE-588)12144371X |4 aut | |
| 240 | 1 | 0 | |a Čislennoe integrirovanie stochastičeskich differencial'nych uravnenij |
| 245 | 1 | 0 | |a Numerical integration of stochastic differential equations |c by G. N. Milstein |
| 264 | 1 | |a Dordrecht u.a. |b Kluwer |c 1995 | |
| 300 | |a VII, 169 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematics and its applications |v 313 | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
Introduction 1
Chapter 1. Mean square approximation of solutions of systems of
stochastic differential equations 11
1. Theorem on the order of convergence (theorem on the relation be¬
tween approximation on a finite interval and one step approximation) 11
1.1. Statement of the theorem 11
1.2. Lemmas 12
1.3. Proof of Theorem 1.1 15
1.4. Discussion 16
1.5. Equations in the sense of Stratonovich 17
1.6. Euler s method 18
1.7. Examples 21
2. Methods based on an analog of Taylor expansion of the solution 23
2.1. Taylor expansion of the solution for systems of ordinary differential
equations 23
2.2. Expansion of the solution of a system of stochastic differential equations
(Wagner Platen expansion) 25
2.3. Construction of implicit methods 35
3. Explicit and implicit methods of order 3/2 for systems with additive
noises 37
3.1. Explicit methods based on Taylor type expansion 37
3.2. Implicit methods based on Taylor type expansion 41
3.3. Stiff systems of stochastic differential equations with additive noises.
.A stability 45
3.4. Runge Kutta type methods (implicit and explicit) 49
3.5. Two step difference methods 52
4. Optimal integration methods for linear systems with additive
noises 56
4.1. Statement of the problem on numerical modeling of the Kalman Bucy
filter and on the optimal filter with discrete arrival of information 57
vi CONTENTS
4.2. Discretisation of the system (4.1), (4.2) 59
4.3. An optimal filter with discrete arrival of information 60
4.4. An optimal integration method of the first order of accuracy 61
5. A strengthening of the main convergence theorem 63
5.1. The theorem on convergence in the mean of order 4 63
5.2. Construction of an auxiliary submartingale 70
5.3. The strenghtened convergence theorem 72
Chapter 2. Modeling of Ito integrals 75
6. Modeling Ito integrals depending on a single noise 75
6.1. Auxiliary formulas for single Ito integrals 76
6.2. Reduction of repeated Ito integrals to single Ito integrals 79
6.3. Exact modeling of the random variables w(h), $w(6)dd, andfoW2(ff)d6 82
6.4. Approximate modeling of the random variables w(h), Jo w{0) d0, and
S£w2{0)d6 84
7. Modeling Ito integrals depending on several noises 90
7.1. Exact methods for modeling the random variables in a method of order
1 in the case of two noises 90
7.2. Use of the numerical integration of special linear stochastic systems for
modeling Ito integrals 91
7.3. Modeling the Ito integrals foWi(s)dwj(s), i,j = 1,... ,q 92
Chapter 3. Weak approximation of solutions of systems of stochastic
differential equations 101
8. One step approximation 101
8.1. Initial assumptions and notations. Lemmas on properties of remainders
and Ito integrals 101
8.2. Forming one step approximations of third order of accuracy 106
8.3. Theorem on a method with one step approximation of third order of
accuracy 109
8.4. Modeling of random variables and constructive formation of a one step
approximation of third order of accuracy 110
9. The main theorem on convergence of weak approximations and
methods of order of accuracy two 112
9.1. A theorem on the relation between one step approximation and ap¬
proximation on a finite interval 112
9.2. Theorem on a method of order of accuracy two 115
9.3. Runge Kutta type methods 116
10. A method of order of accuracy three for systems with additive
noises 118
10.1. Main lemmas 119
10.2. Construction of a one step approximation of order of accuracy four,
and of a method of order three 122
CONTENTS Vii
11. An implicit method 127
12. Reducing the error of the Monte Carlo method 130
Chapter 4. Application of the numerical integration of stochastic equa¬
tions for the Monte Carlo computation of Wiener integrals 135
13. Methods of order of accuracy two for computing Wiener integrals
of functionals of integral type 135
13.1. Statement of the problem 135
13.2. Taylor expansions of mathematical expectations 136
13.3. The trapezium method 138
13.4. The rectangle method and other methods 140
13.5. Generalisation of the trapezium formula to Wiener integrals of func¬
tionals of general form 141
14. Methods of order of accuracy four for computing Wiener integrals
of functionals of exponential type 145
14.1. Introduction 145
14.2. A fourth order Runge Kutta method for integrating the system (14.2) 147
14.3. Reducing variances 152
14.4. Examples of numerical experiments 156
Bibliography 165
Index 169
|
| any_adam_object | 1 |
| author | Milʹstejn, Grigorij N. 1937- |
| author_GND | (DE-588)12144371X |
| author_facet | Milʹstejn, Grigorij N. 1937- |
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| author_sort | Milʹstejn, Grigorij N. 1937- |
| author_variant | g n m gn gnm |
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| ctrlnum | (OCoLC)844947688 (DE-599)BVBBV010276444 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV010276444 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:49:42Z |
| institution | BVB |
| isbn | 079233213X |
| language | English |
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| physical | VII, 169 S. |
| publishDate | 1995 |
| publishDateSearch | 1995 |
| publishDateSort | 1995 |
| publisher | Kluwer |
| record_format | marc |
| series | Mathematics and its applications |
| series2 | Mathematics and its applications |
| spelling | Milʹstejn, Grigorij N. 1937- Verfasser (DE-588)12144371X aut Čislennoe integrirovanie stochastičeskich differencial'nych uravnenij Numerical integration of stochastic differential equations by G. N. Milstein Dordrecht u.a. Kluwer 1995 VII, 169 S. txt rdacontent n rdamedia nc rdacarrier Mathematics and its applications 313 Aus dem Russ. übers. Stochastische Differentialgleichung (DE-588)4057621-8 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Stochastische Differentialgleichung (DE-588)4057621-8 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Mathematics and its applications 313 (DE-604)BV008163334 313 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006837186&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Milʹstejn, Grigorij N. 1937- Numerical integration of stochastic differential equations Mathematics and its applications Stochastische Differentialgleichung (DE-588)4057621-8 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| subject_GND | (DE-588)4057621-8 (DE-588)4128130-5 |
| title | Numerical integration of stochastic differential equations |
| title_alt | Čislennoe integrirovanie stochastičeskich differencial'nych uravnenij |
| title_auth | Numerical integration of stochastic differential equations |
| title_exact_search | Numerical integration of stochastic differential equations |
| title_full | Numerical integration of stochastic differential equations by G. N. Milstein |
| title_fullStr | Numerical integration of stochastic differential equations by G. N. Milstein |
| title_full_unstemmed | Numerical integration of stochastic differential equations by G. N. Milstein |
| title_short | Numerical integration of stochastic differential equations |
| title_sort | numerical integration of stochastic differential equations |
| topic | Stochastische Differentialgleichung (DE-588)4057621-8 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| topic_facet | Stochastische Differentialgleichung Numerisches Verfahren |
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