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

Seiichiro Kusuoka

    Research output: Contribution to journalArticlepeer-review

    7 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

    Keywords

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

    ASJC Scopus subject areas

    • Analysis

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