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

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. © 2009 Elsevier Inc. All rights reserved.