Martingale Representation and the Malliavin Calculus.

Abstract

Using the theory of stochastic flows the integrand in a stochastic integral is identified. After some rearrangement this integrand is itself written in terms of a martingale which can be expressed as a stochastic integral, and by recursively repeating the representation a homogeneous chaos expansion is obtained. Using the stochastic integral representation an integration by parts formula is then derived. If the inverse of the Malliavin matrix M belongs to all the spaces L superscript p (Omega) we show a random variable has a smooth density. The difficult questions concerning the relationship between Hoermander's conditions on the coefficient vector fields and the integrability of 1/M are not discussed, but, at least for Markov flows, the discussion below appears to be an elementary treatment of some ideas of the Malliavin calculus.

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Document Details

Document Type
Technical Report
Publication Date
Nov 11, 1987
Accession Number
ADA189721

Entities

People

  • Michael Kohlmann
  • Robert J. Elliott

Organizations

  • University of Alberta

Tags

Communities of Interest

  • Materials and Manufacturing Processes

DTIC Thesaurus Topics

  • Air Force
  • Calculus
  • Classification
  • Coefficients
  • Differential Equations
  • Equations
  • Filtration
  • Integrals
  • Kolmogorov Equations
  • Mathematical Analysis
  • New York
  • Partial Differential Equations
  • Probability
  • Random Variables
  • United States
  • Universities

Fields of Study

  • Mathematics

Readers

  • Calculus or Mathematical Analysis
  • Mathematical Modeling and Probability Theory.

Technology Areas

  • Space