skip to content
Brownian motion and stochastic calculus Preview this item
ClosePreview this item

Brownian motion and stochastic calculus

Author: Ioannis Karatzas; Steven E Shreve
Publisher: New York : Springer-Verlag, ©1991.
Series: Graduate texts in mathematics, 113.
Edition/Format:   Print book : English : 2nd edView all editions and formats

A graduate-course text, written for readers familiar with measure-theoretic probability and discrete-time processes, wishing to explore stochastic processes in continuous time.


(not yet rated) 0 with reviews - Be the first.

More like this

Find a copy online

Links to this item

Find a copy in the library

&AllPage.SpinnerRetrieving; Finding libraries that hold this item...


Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Ioannis Karatzas; Steven E Shreve
ISBN: 0387976558 9780387976556 3540976558 9783540976554
OCLC Number: 24011122
Notes: "Springer study edition"--Cover.
Description: xxiii, 470 pages : illustrations ; 24 cm.
Contents: 1 Martingales, Stopping Times, and Filtrations.- 1.1. Stochastic Processes and ?-Fields.- 1.2. Stopping Times.- 1.3. Continuous-Time Martingales.- A. Fundamental inequalities.- B. Convergence results.- C. The optional sampling theorem.- 1.4. The Doob-Meyer Decomposition.- 1.5. Continuous, Square-Integrable Martingales.- 1.6. Solutions to Selected Problems.- 1.7. Notes.- 2 Brownian Motion.- 2.1. Introduction.- 2.2. First Construction of Brownian Motion.- A. The consistency theorem.- B. The Kolmogorov-?entsov theorem.- 2.3. Second Construction of Brownian Motion.- 2.4. The SpaceC[0, ?), Weak Convergence, and Wiener Measure.- A. Weak convergence.- B. Tightness.- C. Convergence of finite-dimensional distributions.- D. The invariance principle and the Wiener measure.- 2.5. The Markov Property.- A. Brownian motion in several dimensions.- B. Markov processes and Markov families.- C. Equivalent formulations of the Markov property.- 2.6. The Strong Markov Property and the Reflection Principle.- A. The reflection principle.- B. Strong Markov processes and families.- C. The strong Markov property for Brownian motion.- 2.7. Brownian Filtrations.- A. Right-continuity of the augmented filtration for a strong Markov process.- B. A "universal" filtration.- C. The Blumenthal zero-one law.- 2.8. Computations Based on Passage Times.- A. Brownian motion and its running maximum.- B. Brownian motion on a half-line.- C. Brownian motion on a finite interval.- D. Distributions involving last exit times.- 2.9. The Brownian Sample Paths.- A. Elementary properties.- B. The zero set and the quadratic variation.- C. Local maxima and points of increase.- D. Nowhere differentiability.- E. Law of the iterated logarithm.- F. Modulus of continuity.- 2.10. Solutions to Selected Problems.- 2.11. Notes.- 3 Stochastic Integration.- 3.1. Introduction.- 3.2. Construction of the Stochastic Integral.- A. Simple processes and approximations.- B. Construction and elementary properties of the integral.- C. A characterization of the integral.- D. Integration with respect to continuous, local martingales.- 3.3. The Change-of-Variable Formula.- A. The Ito rule.- B. Martingale characterization of Brownian motion.- C. Bessel processes, questions of recurrence.- D. Martingale moment inequalities.- E. Supplementary exercises.- 3.4. Representations of Continuous Martingales in Terms of Brownian Motion.- A. Continuous local martingales as stochastic integrals with respect to Brownian motion.- B. Continuous local martingales as time-changed Brownian motions.- C. A theorem of F. B. Knight.- D. Brownian martingales as stochastic integrals.- E. Brownian functionals as stochastic integrals.- 3.5. The Girsanov Theorem.- A. The basic result.- B. Proof and ramifications.- C. Brownian motion with drift.- D. The Novikov condition.- 3.6. Local Time and a Generalized Ito Rule for Brownian Motion.- A. Definition of local time and the Tanaka formula.- B. The Trotter existence theorem.- C. Reflected Brownian motion and the Skorohod equation.- D. A generalized Ito rule for convex functions.- E. The Engelbert-Schmidt zero-one law.- 3.7. Local Time for Continuous Semimartingales.- 3.8. Solutions to Selected Problems.- 3.9. Notes.- 4 Brownian Motion and Partial Differential Equations.- 4.1. Introduction.- 4.2. Harmonic Functions and the Dirichlet Problem.- A. The mean-value property.- B. The Dirichlet problem.- C. Conditions for regularity.- D. Integral formulas of Poisson.- E. Supplementary exercises.- 4.3. The One-Dimensional Heat Equation.- A. The Tychonoff uniqueness theorem.- B. Nonnegative solutions of the heat equation.- C. Boundary crossing probabilities for Brownian motion.- D. Mixed initial/boundary value problems.- 4.4. The Formulas of Feynman and Kac.- A. The multidimensional formula.- B. The one-dimensional formula.- 4.5. Solutions to selected problems.- 4.6. Notes.- 5 Stochastic Differential Equations.- 5.1. Introduction.- 5.2. Strong Solutions.- A. Definitions.- B. The Ito theory.- C. Comparison results and other refinements.- D. Approximations of stochastic differential equations.- E. Supplementary exercises.- 5.3. Weak Solutions.- A. Two notions of uniqueness.- B. Weak solutions by means of the Girsanov theorem.- C. A digression on regular conditional probabilities.- D. Results of Yamada and Watanabe on weak and strong solutions.- 5.4. The Martingale Problem of Stroock and Varadhan.- A. Some fundamental martingales.- B. Weak solutions and martingale problems.- C. Well-posedness and the strong Markov property.- D. Questions of existence.- E. Questions of uniqueness.- F. Supplementary exercises.- 5.5. A Study of the One-Dimensional Case.- A. The method of time change.- B. The method of removal of drift.- C. Feller's test for explosions.- D. Supplementary exercises.- 5.6. Linear Equations.- A. Gauss-Markov processes.- B. Brownian bridge.- C. The general, one-dimensional, linear equation.- D. Supplementary exercises.- 5.7. Connections with Partial Differential Equations.- A. The Dirichlet problem.- B. The Cauchy problem and a Feynman-Kac representation.- C. Supplementary exercises.- 5.8. Applications to Economics.- A. Portfolio and consumption processes.- B. Option pricing.- C. Optimal consumption and investment (general theory).- D. Optimal consumption and investment (constant coefficients).- 5.9. Solutions to Selected Problems.- 5.10. Notes.- 6 P. Levy's Theory of Brownian Local Time.- 6.1. Introduction.- 6.2. Alternate Representations of Brownian Local Time.- A. The process of passage times.- B. Poisson random measures.- C. Subordinators.- D. The process of passage times revisited.- E. The excursion and downcrossing representations of local time.- 6.3. Two Independent Reflected Brownian Motions.- A. The positive and negative parts of a Brownian motion.- B. The first formula of D. Williams.- C. The joint density of (W(t), L(t), ?+(t)).- 6.4. Elastic Brownian Motion.- A. The Feynman-Kac formulas for elastic Brownian motion.- B. The Ray-Knight description of local time.- C. The second formula of D. Williams.- 6.5. An Application: Transition Probabilities of Brownian Motion with Two-Valued Drift.- 6.6. Solutions to Selected Problems.- 6.7. Notes.
Series Title: Graduate texts in mathematics, 113.
Responsibility: Ioannis Karatzas, Steven E. Shreve.
More information:


Editorial reviews

Publisher Synopsis

Second EditionI. Karatzas and S.E. ShreveBrownian Motion and Stochastic Calculus"A valuable book for every graduate student studying stochastic process, and for those who are interested in pure and Read more...

User-contributed reviews
Retrieving GoodReads reviews...
Retrieving DOGObooks reviews...


Be the first.
Confirm this request

You may have already requested this item. Please select Ok if you would like to proceed with this request anyway.

Linked Data

Primary Entity

<> # Brownian motion and stochastic calculus
    a schema:Book, schema:CreativeWork ;
    library:oclcnum "24011122" ;
    library:placeOfPublication <> ; # New York
    library:placeOfPublication <> ;
    schema:about <> ;
    schema:about <> ; # Stochastic analysis
    schema:about <> ;
    schema:about <> ; # Stochastische analyse
    schema:about <> ; # Brownian motion processes
    schema:about <> ; # Brownse beweging
    schema:about <> ; # Análisis estocástico
    schema:about <> ; # Analyse stochastique
    schema:about <> ; # Mouvement brownien, Processus de
    schema:bookEdition "2nd ed." ;
    schema:bookFormat bgn:PrintBook ;
    schema:contributor <> ; # Steven E. Shreve
    schema:copyrightYear "1991" ;
    schema:creator <> ; # Ioannis Karatzas
    schema:datePublished "1991" ;
    schema:exampleOfWork <> ;
    schema:inLanguage "en" ;
    schema:isPartOf <> ; # Graduate texts in mathematics ;
    schema:name "Brownian motion and stochastic calculus"@en ;
    schema:productID "24011122" ;
    schema:publication <> ;
    schema:publisher <> ; # Springer-Verlag
    schema:url <> ;
    schema:url <> ;
    schema:workExample <> ;
    schema:workExample <> ;
    umbel:isLike <> ;
    wdrs:describedby <> ;

Related Entities

<> # New York
    a schema:Place ;
    schema:name "New York" ;

<> # Springer-Verlag
    a bgn:Agent ;
    schema:name "Springer-Verlag" ;

<> # Graduate texts in mathematics ;
    a bgn:PublicationSeries ;
    schema:hasPart <> ; # Brownian motion and stochastic calculus
    schema:name "Graduate texts in mathematics ;" ;

<> # Análisis estocástico
    a schema:Intangible ;
    schema:name "Análisis estocástico"@en ;

<> # Analyse stochastique
    a schema:Intangible ;
    schema:name "Analyse stochastique"@fr ;

<> # Brownse beweging
    a schema:Intangible ;
    schema:name "Brownse beweging"@en ;

<> # Mouvement brownien, Processus de
    a schema:Intangible ;
    schema:name "Mouvement brownien, Processus de"@fr ;

<> # Stochastische analyse
    a schema:Intangible ;
    schema:name "Stochastische analyse"@en ;

<> # Stochastic analysis
    a schema:Intangible ;
    schema:name "Stochastic analysis"@en ;

<> # Brownian motion processes
    a schema:Intangible ;
    schema:name "Brownian motion processes"@en ;

<> # Steven E. Shreve
    a schema:Person ;
    schema:familyName "Shreve" ;
    schema:givenName "Steven E." ;
    schema:name "Steven E. Shreve" ;

<> # Ioannis Karatzas
    a schema:Person ;
    schema:familyName "Karatzas" ;
    schema:givenName "Ioannis" ;
    schema:name "Ioannis Karatzas" ;

    a schema:ProductModel ;
    schema:isbn "0387976558" ;
    schema:isbn "9780387976556" ;

    a schema:ProductModel ;
    schema:isbn "3540976558" ;
    schema:isbn "9783540976554" ;

Content-negotiable representations

Close Window

Please sign in to WorldCat 

Don't have an account? You can easily create a free account.