Existence of densities of solutions of stochastic differential equations by Malliavin calculus

Seiichiro Kusuoka

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

I considered if solutions of stochastic differential equations have their density or not when the coefficients are not Lipschitz continuous. However, when stochastic differential equations whose coefficients are not Lipschitz continuous, the solutions would not belong to Sobolev space in general. So, I prepared the class Vh which is larger than Sobolev space, and considered the relation between absolute continuity of random variables and the class Vh. The relation is associated to a theorem of N. Bouleau and F. Hirsch. Moreover, I got a sufficient condition for a solution of stochastic differential equation to belong to the class Vh, and showed that solutions of stochastic differential equations have their densities in a special case by using the class Vh.

Original languageEnglish
Pages (from-to)758-784
Number of pages27
JournalJournal of Functional Analysis
Volume258
Issue number3
DOIs
Publication statusPublished - Feb 1 2010
Externally publishedYes

Fingerprint

Malliavin Calculus
Stochastic Equations
Differential equation
Sobolev Spaces
Lipschitz
Absolute Continuity
Coefficient
Random variable
Class
Sufficient Conditions
Theorem

Keywords

  • Absolute continuity
  • Existence of densities
  • Existence of fundamental solutions
  • Malliavin calculus
  • Stochastic differential equation

ASJC Scopus subject areas

  • Analysis

Cite this

Existence of densities of solutions of stochastic differential equations by Malliavin calculus. / Kusuoka, Seiichiro.

In: Journal of Functional Analysis, Vol. 258, No. 3, 01.02.2010, p. 758-784.

Research output: Contribution to journalArticle

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