Accession Number : ADA189721

Title :   Martingale Representation and the Malliavin Calculus.

Descriptive Note : Rept. for 30 Sep 86-30 Sep 87,

Corporate Author : ALBERTA UNIV EDMONTON DEPT OF STATISTICS AND APPLIED PROBABILITY

Personal Author(s) : Elliott, Robert J ; Kohlmann, Michael

PDF Url : ADA189721

Report Date : 11 Nov 1987

Pagination or Media Count : 13

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.

Descriptors :   *STOCHASTIC PROCESSES, *MATHEMATICAL FILTERS, CALCULUS, COEFFICIENTS, FORMULATIONS, INTEGRALS, INTEGRATION, THEORY, VECTOR ANALYSIS, BROWNIAN MOTION

Subject Categories : Statistics and Probability

Distribution Statement : APPROVED FOR PUBLIC RELEASE