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

• Published in: Journal of functional analysis Vol. 258; no. 3; pp. 758 - 784
• Main Author: Kusuoka, Seiichiro
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.02.2010
• Subjects: Absolute continuity; Existence of densities; Existence of fundamental solutions; Malliavin calculus; Stochastic differential equation; Absolute continuity; Existence of fundamental solutions; Existence of densities; Malliavin calculus; Stochastic differential equation
• ISSN: 0022-1236; 1096-0783
• DOI: 10.1016/j.jfa.2009.09.009

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 V h which is larger than Sobolev space, and considered the relation between absolute continuity of random variables and the class V h . 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 V h , and showed that solutions of stochastic differential equations have their densities in a special case by using the class V h .