I. Karatzas, S.E.Shreve: Brownian Motion and Stochastic Calculus

Title: Brownian Motion and Stochastic Calculus
Author: Ioannis Karatzas, Steven E.Shreve
latest edition: 2nd edition, Springer New York (1991)
ISBN: 0387976558

Description

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, which is presented as the canonical example of both a martingale and a Markov process with continuous paths. In this context, the theory of stochastic integration and stochastic calculus is developed. The power of this calculus is illustrated by results concerning representations of martingales and change of measure on Wiener space, and these in turn permit a presentation of recent advances in financial economics (option pricing and consumption/investment optimization). This book contains a detailed discussion of weak and strong solutions of stochastic differential equations and a study of local time for semimartingales, with special emphasis on the theory of Brownian local time. The text is complemented by a large number of problems and exercises.

Table of contents

  2. Martingales, Stopping times, and Filtrations
    1. Stochastic processes and σ-fields
    2. Stopping times
    3. Continuous-time martingales
      1. Fundamental inequalities
      2. Convergence results
      3. The optional sampling theorem
    4. The Doob-Meyer decomposition
    5. Continuous and square-integrable martingales
    6. Solutions to selected problems
    7. Notes
  3. Brownian motion
    1. Introduction
    2. First construction of Brownian motion
      1. The consistency theorem
      2. The Kolmogorov-Chentsov theorem
    3. Second construction of Brownian motion
    4. The space C[0,∞), weak convergence, and Wiener measure
      1. Weak convergence
      2. Tightness
      3. Convergence of finite-dimensional distributions
      4. The invariance principle and the Wiener measure
    5. The Markov property
      1. Brownian motion in several dimensions
      2. Markov processes and Markov families
      3. Equivalent formulations of the Markov property
    6. The strong Markov property and the reflection principle
      1. The reflection principle
      2. Strong Markov processes and families
      3. The strong Markov property for Brownian motion
    7. Brownian filtrations
      1. Right-continuity of the augmented filtration for a strong Markov process
      2. A "universal" filtration
      3. The Blumenthal zero-one law
    8. Computations based on passage times
      1. Brownian motion and its running maximum
      2. Brownian motion on a half-line
      3. Brownian motion on a finite interval
      4. Distributions involving last exit times
    9. Brownian sample paths
      1. Elementary properties
      2. The zero set and the quadratic variation
      3. Local maxima and points of increase
      4. Nowhere differentiability
      5. Law of the iterated logarithm
      6. Modulus of continuity
    10. Solutions to selected problems
    11. Notes
  4. Stochastic integration
    1. Introduction
    2. Construction of the stochastic integral
      1. Simple processes and approximation
      2. Construction and elementary properties of the integral
      3. A characterization of the integral
      4. Integration with respect to continuous, local martingales
    3. Change-of-variable formula
      1. The Itô-rule
      2. Martingale characterization of Brownian motion
      3. Bessel processes, questions of recurrence
      4. Martingale moment inequalities
      5. Supplementary exercises
    4. Representations of continuous martingales in terms of Brownian motion
      1. Continuous local martingales as stochastic integrals with respect to Brownian motion
      2. Coninuous local martingales as time-changed Brownian motions
      3. A theorem of F.B.Knight
      4. Brownian martingales as stochastic integrals
      5. Brownian functionals as stochastic integrals
    5. Grisanov theorem
      1. The basic result
      2. Proof and ramifications
      3. Brownian motion with drift
      4. The Novikov condition
    6. Local time and generalized Itô rule for Brownian motion
      1. Definition of local time and the Tanaka formula
      2. The Trotter existence theorem
      3. Reflected Brownian motion and the Skorohod equation
      4. A generalized Itô rule for convex functions
      5. The Engelbert-Schmidt zero-one law
    7. Local time for continuous semimartingales
    8. Solutions to selected problems
    9. Notes
  5. Brownian motion and Partial differential equations
    1. Introduction
    2. Harmonic functions and the Dirichlet problem
      1. …
    3. …
  6. Stochastic differential equations
    1. Introduction
    2. Strong solutions
      1. Definitions
      2. The Itô theory
      3. Comparison results and other refinements
      4. Approximations of stochastic differential equations
      5. Supplementary exercises
    3. Weak solutions
      1. Two notions of uniqueness
      2. Weak solutions by means of the Grisanov theorem
      3. Digression on regular conditional probabilities
      4. Results of Yamada and Watanabe on weak and strong solutions
    4. The martingale problem of Stroock and Varadhan
      1. Some fundamental martingales
      2. Weak solutions and martingale problems
      3. Well-posedness and the strong Markov property
      4. Questions of existence
      5. Questions of uniqueness
      6. Supplementary exercises
    5. Study of the one-dimensional case
      1. The method of time change
      2. The method of removal of drift
      3. Feller's test for explosions
      4. Supplementary exercises
    6. Linear equations
      1. Gauss-Markov process
      2. Brownian bridge
      3. …
    7. Connections with partial differential equations
    8. Applications to economics
      1. Portfolio and consumption processes
      2. Option pricing
      3. Optimal consumption and investment I (general theory)
      4. Optimal consumption and investment II (constant coefficients)
    9. Solutions to selected problems
    10. Notes
  7. Lèvy's theory of Brownian local time
    1. …
  8. Bibliography
  9. Index