MARC details
| 000 -LEADER |
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| fixed length control field |
02332 a2200229 4500 |
| 008 - FIXED-LENGTH DATA ELEMENTS--GENERAL INFORMATION |
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230127b |||||||| |||| 00| 0 eng d |
| 020 ## - INTERNATIONAL STANDARD BOOK NUMBER |
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| International Standard Book Number |
9780387287201 |
| 082 ## - DEWEY DECIMAL CLASSIFICATION NUMBER |
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| Classification number |
519.23 |
| Item number |
KUO |
| 100 ## - MAIN ENTRY--PERSONAL NAME |
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| Personal name |
Kuo, Hui-Hsiung |
| 245 ## - TITLE STATEMENT |
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| Title |
Introduction to stochastic integration |
| 260 ## - PUBLICATION, DISTRIBUTION, ETC. (IMPRINT) |
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| Name of publisher, distributor, etc |
Springer Science + Business media, |
| Date of publication, distribution, etc |
2006. |
| Place of publication, distribution, etc |
New York: |
| 300 ## - PHYSICAL DESCRIPTION |
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| Extent |
xiii, 278p.; |
| Other physical details |
pbk; |
| Dimensions |
23cm. |
| 440 ## - SERIES STATEMENT/ADDED ENTRY--TITLE |
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| Title |
Universitext |
| 504 ## - BIBLIOGRAPHY, ETC. NOTE |
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| Bibliography, etc |
Include Index and Reference |
| 520 ## - SUMMARY, ETC. |
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| Summary, etc |
In the Leibniz–Newton calculus, one learns the di?erentiation and integration of deterministic functions. A basic theorem in di?erentiation is the chain rule, which gives the derivative of a composite of two di?erentiable functions. The chain rule, when written in an inde?nite integral form, yields the method of substitution. In advanced calculus, the Riemann–Stieltjes integral is de?ned through the same procedure of “partition-evaluation-summation-limit” as in the Riemann integral. In dealing with random functions such as functions of a Brownian motion, the chain rule for the Leibniz–Newton calculus breaks down. A Brownian motionmovessorapidlyandirregularlythatalmostallofitssamplepathsare nowhere di?erentiable. Thus we cannot di?erentiate functions of a Brownian motion in the same way as in the Leibniz–Newton calculus. In 1944 Kiyosi Itˆ o published the celebrated paper “Stochastic Integral” in the Proceedings of the Imperial Academy (Tokyo). It was the beginning of the Itˆ o calculus, the counterpart of the Leibniz–Newton calculus for random functions. In this six-page paper, Itˆ o introduced the stochastic integral and a formula, known since then as Itˆ o’s formula. The Itˆ o formula is the chain rule for the Itˆocalculus.Butitcannotbe expressed as in the Leibniz–Newton calculus in terms of derivatives, since a Brownian motion path is nowhere di?erentiable. The Itˆ o formula can be interpreted only in the integral form. Moreover, there is an additional term in the formula, called the Itˆ o correction term, resulting from the nonzero quadratic variation of a Brownian motion.<br/><br/><br/>https://link.springer.com/book/10.1007/0-387-31057-6 |
| 650 ## - SUBJECT ADDED ENTRY--TOPICAL TERM |
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| Topical term or geographic name as entry element |
Martingale |
| 650 ## - SUBJECT ADDED ENTRY--TOPICAL TERM |
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| Topical term or geographic name as entry element |
Gaussian measure |
| 650 ## - SUBJECT ADDED ENTRY--TOPICAL TERM |
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| Topical term or geographic name as entry element |
Brownian motion |
| 650 ## - SUBJECT ADDED ENTRY--TOPICAL TERM |
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| Topical term or geographic name as entry element |
Probability theory |
| 650 ## - SUBJECT ADDED ENTRY--TOPICAL TERM |
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| Topical term or geographic name as entry element |
Diffusion process |
| 942 ## - ADDED ENTRY ELEMENTS (KOHA) |
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| Source of classification or shelving scheme |
Dewey Decimal Classification |
| Item type |
Books |
