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Brownian motion and stochastic calculus /

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This book is designed as a text for graduate courses in stochastic processes. It is written for readers familiar with measure-theoretic probability and discrete-time processes who wish to explore stochastic processes in continuous time. The vehicle chosen for this exposition is Brownian motion, whic...

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Bibliographic Details
Main Authors: Karatzas, Ioannis (Author), Shreve, Steven E. (Author)
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: New York : Springer, 1996.
Edition:Second edition.
Series:Graduate texts in mathematics ; 113.
Subjects:
Brownian motion processes.
Stochastic analysis.
Brownsche Bewegung.
Stochastische Analysis.
Electronic books.
Online Access:Connect to the full text of this electronic book
Table of Contents:
  • 1 Martingales, Stopping Times, and Filtrations
  • 1.1. Stochastic Processes and ?-Fields
  • 1.2. Stopping Times
  • 1.3. Continuous-Time Martingales
  • 1.4. The DoobMeyer 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
  • 2.3. Second Construction of Brownian Motion
  • 2.4. The SpaceC[0, ?), Weak Convergence, and Wiener Measure
  • 2.5. The Markov Property
  • 2.6. The Strong Markov Property and the Reflection Principle
  • 2.7. Brownian Filtrations
  • 2.8. Computations Based on Passage Times
  • 2.9. The Brownian Sample Paths
  • 2.10. Solutions to Selected Problems
  • 2.11. Notes
  • 3 Stochastic Integration
  • 3.1. Introduction
  • 3.2. Construction of the Stochastic Integral
  • 3.3. The Change-of-Variable Formula
  • 3.4. Representations of Continuous Martingales in Terms of Brownian Motion
  • 3.5. The Girsanov Theorem
  • 3.6. Local Time and a Generalized It Rule for Brownian Motion
  • 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
  • 4.3. The One-Dimensional Heat Equation
  • 4.4. The Formulas of Feynman and Kac
  • 4.5. Solutions to selected problems
  • 4.6. Notes
  • 5 Stochastic Differential Equations
  • 5.1. Introduction
  • 5.2. Strong Solutions
  • 5.3. Weak Solutions
  • 5.4. The Martingale Problem of Stroock and Varadhan
  • 5.5. A Study of the One-Dimensional Case
  • 5.6. Linear Equations
  • 5.7. Connections with Partial Differential Equations
  • 5.8. Applications to Economics
  • 5.9. Solutions to Selected Problems
  • 5.10. Notes
  • 6 P. Lvys Theory of Brownian Local Time
  • 6.1. Introduction
  • 6.2. Alternate Representations of Brownian Local Time
  • 6.3. Two Independent Reflected Brownian Motions
  • 6.4. Elastic Brownian Motion
  • 6.5. An Application: Transition Probabilities of Brownian Motion with Two-Valued Drift
  • 6.6. Solutions to Selected Problems
  • 6.7. Notes.