1st Edition

Stochastic Calculus A Practical Introduction

By Richard Durrett Copyright 1996

    This compact yet thorough text zeros in on the parts of the theory that are particularly relevant to applications . It begins with a description of Brownian motion and the associated stochastic calculus, including their relationship to partial differential equations. It solves stochastic differential equations by a variety of methods and studies in detail the one-dimensional case. The book concludes with a treatment of semigroups and generators, applying the theory of Harris chains to diffusions, and presenting a quick course in weak convergence of Markov chains to diffusions.

    The presentation is unparalleled in its clarity and simplicity. Whether your students are interested in probability, analysis, differential geometry or applications in operations research, physics, finance, or the many other areas to which the subject applies, you'll find that this text brings together the material you need to effectively and efficiently impart the practical background they need.

    CHAPTER 1. BROWNIAN MOTION
    Definition and Construction
    Markov Property, Blumenthal's 0-1 Law
    Stopping Times, Strong Markov Property
    First Formulas
    CHAPTER 2. STOCHASTIC INTEGRATION
    Integrands: Predictable Processes
    Integrators: Continuous Local Martingales
    Variance and Covariance Processes
    Integration w.r.t. Bounded Martingales
    The Kunita-Watanabe Inequality
    Integration w.r.t. Local Martingales
    Change of Variables, Itô's Formula
    Integration w.r.t. Semimartingales
    Associative Law
    Functions of Several Semimartingales
    Chapter Summary
    Meyer-Tanaka Formula, Local Time
    Girsanov's Formula
    CHAPTER 3. BROWNIAN MOTION, II
    Recurrence and Transience
    Occupation Times
    Exit Times
    Change of Time, Lévy's Theorem
    Burkholder Davis Gundy Inequalities
    Martingales Adapted to Brownian Filtrations
    CHAPTER 4. PARTIAL DIFFERENTIAL EQUATIONS
    A. Parabolic Equations
    The Heat Equation
    The Inhomogeneous Equation
    The Feynman-Kac Formula
    B. Elliptic Equations
    The Dirichlet Problem
    Poisson's Equation
    The Schrödinger Equation
    C. Applications to Brownian Motion
    Exit Distributions for the Ball
    Occupation Times for the Ball
    Laplace Transforms, Arcsine Law
    CHAPTER 5. STOCHASTIC DIFFERENTIAL EQUATIONS
    Examples
    Itô's Approach
    Extension
    Weak Solutions
    Change of Measure
    Change of Time
    CHAPTER 6. ONE DIMENSIONAL DIFFUSIONS
    Construction
    Feller's Test
    Recurrence and Transience
    Green's Functions
    Boundary Behavior
    Applications to Higher Dimensions
    CHAPTER 7. DIFFUSIONS AS MARKOV PROCESSES
    Semigroups and Generators
    Examples
    Transition Probabilities
    Harris Chains
    Convergence Theorems
    CHAPTER 8. WEAK CONVERGENCE
    In Metric Spaces
    Prokhorov's Theorems
    The Space C
    Skorohod's Existence Theorem for SDE
    Donsker's Theorem
    The Space D
    Convergence to Diffusions
    Examples
    Solutions to Exercises
    References
    Index

    Biography

    Richard Durrett