Brownian motion calculus
 Author/Creator
 Wiersema, Ubbo F.
 Language
 English.
 Imprint
 Chichester, England ; Hoboken, NJ : John Wiley & Sons, c2008.
 Physical description
 xv, 313 p. : ill ; 23 cm.
 Series
 Wiley finance series.
Access
Contents/Summary
 Bibliography
 Includes bibliographical references (p. [299]302) and index.
 Contents

 Preface. 1 Brownian Motion. 1.1 Origins. 1.2 Brownian Motion Specification. 1.3 Use of Brownian Motion in Stock Price Dynamics. 1.4 Construction of Brownian Motion from a Symmetric Random Walk. 1.5 Covariance of Brownian Motion. 1.6 Correlated Brownian Motions. 1.7 Successive Brownian Motion Increments. 1.8 Features of a Brownian Motion Path. 1.9 Exercises. 1.10 Summary. 2 Martingales. 2.1 Simple Example. 2.2 Filtration. 2.3 Conditional Expectation. 2.4 Martingale Description. 2.5 Martingale Analysis Steps. 2.6 Examples of Martingale Analysis. 2.7 Process of Independent Increments. 2.8 Exercises. 2.9 Summary. 3 Ito Stochastic Integral. 3.1 How a Stochastic Integral Arises. 3.2 Stochastic Integral for NonRandom StepFunctions. 3.3 Stochastic Integral for NonAnticipating Random StepFunctions. 3.4 Extension to NonAnticipating General Random Integrands. 3.5 Properties of an Ito Stochastic Integral. 3.6 Significance of Integrand Position. 3.7 Ito integral of NonRandom Integrand. 3.8 Area under a Brownian Motion Path. 3.9 Exercises. 3.10 Summary. 3.11 A Tribute to Kiyosi Ito. Acknowledgment. 4 Ito Calculus. 4.1 Stochastic Differential Notation. 4.2 Taylor Expansion in Ordinary Calculus. 4.3 Ito's Formula as a Set of Rules. 4.4 Illustrations of Ito's Formula. 4.5 Levy Characterization of Brownian Motion. 4.6 Combinations of Brownian Motions. 4.7 Multiple Correlated Brownian Motions. 4.8 Area under a Brownian Motion Path  Revisited. 4.9 Justification of Ito's Formula. 4.10 Exercises. 4.11 Summary. 5 Stochastic Differential Equations. 5.1 Structure of a Stochastic Differential Equation. 5.2 Arithmetic Brownian Motion SDE. 5.3 Geometric Brownian Motion SDE. 5.4 OrnsteinUhlenbeck SDE. 5.5 MeanReversion SDE. 5.6 MeanReversion with SquareRoot Diffusion SDE. 5.7 Expected Value of SquareRoot Diffusion Process. 5.8 Coupled SDEs. 5.9 Checking the Solution of a SDE. 5.10 General Solution Methods for Linear SDEs. 5.11 Martingale Representation. 5.12 Exercises. 5.13 Summary. 6 Option Valuation. 6.1 Partial Differential Equation Method. 6.2 Martingale Method in OnePeriod Binomial Framework. 6.3 Martingale Method in ContinuousTime Framework. 6.4 Overview of RiskNeutral Method. 6.5 Martingale Method Valuation of Some European Options. 6.6 Links between Methods. 6.6.1 FeynmanKac Link between PDE Method and Martingale Method. 6.6.2 MultiPeriod Binomial Link to Continuous. 6.7 Exercise. 6.8 Summary. 7 Change of Probability. 7.1 Change of Discrete Probability Mass. 7.2 Change of Normal Density. 7.3 Change of Brownian Motion. 7.4 Girsanov Transformation. 7.5 Use in Stock Price Dynamics  Revisited. 7.6 General Drift Change. 7.7 Use in Importance Sampling. 7.8 Use in Deriving Conditional Expectations. 7.9 Concept of Change of Probability. 7.10 Exercises. 7.11 Summary. 8 Numeraire. 8.1 Change of Numeraire. 8.2 Forward Price Dynamics. 8.3 Option Valuation under most Suitable Numeraire. 8.4 Relating Change of Numeraire to Change of Probability. 8.5 Change of Numeraire for Geometric Brownian Motion. 8.6 Change of Numeraire in LIBOR Market Model. 8.7 Application in Credit Risk Modelling. 8.8 Exercises. 8.9 Summary. ANNEXES. A Annex A: Computations with Brownian Motion. A.1 Moment Generating Function and Moments of Brownian Motion. A.2 Probability of Brownian Motion Position. A.3 Brownian Motion Reflected at the Origin. A.4 First Passage of a Barrier. A.5 Alternative Brownian Motion Specification. B Annex B: Ordinary Integration. B.1 Riemann Integral. B.2 RiemannStieltjes Integral. B.3 Other Useful Properties. B.4 References. C Annex C: Brownian Motion Variability. C.1 Quadratic Variation. C.2 First Variation. D Annex D: Norms. D.1 Distance between Points. D.2 Norm of a Function. D.3 Norm of a Random Variable. D.4 Norm of a Random Process. D.5 Reference. E Annex E: Convergence Concepts. E.1 Central Limit Theorem. E.2 MeanSquare Convergence. E.3 Almost Sure Convergence. E.4 Convergence in Probability. E.5 Summary. Answers to Exercises. References. Index.
 (source: Nielsen Book Data)
 Publisher's Summary
 Brownian Motion Calculus presents the basics of Stochastic Calculus with a focus on the valuation of financial derivatives. It is intended as an accessible introduction to the technical literature. A clear distinction has been made between the mathematics that is convenient for a first introduction, and the more rigorous underpinnings which are best studied from the selected technical references. The inclusion of fully worked out exercises makes the book attractive for self study. Standard probability theory and ordinary calculus are the prerequisites. Summary slides for revision and teaching can be found on the book website.
(source: Nielsen Book Data)  Supplemental links

Table of contents only
Table of Contents
Table of Contents
Table of Contents Brownian motion calculus
Subjects
Bibliographic information
 Publication date
 2008
 Responsibility
 Ubbo F Wiersema.
 ISBN
 9780470021705 (pbk. : alk. paper)
 0470021705 (pbk. : alk. paper)