### 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 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}.

Original language | English |
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Pages (from-to) | 758-784 |

Number of pages | 27 |

Journal | Journal of Functional Analysis |

Volume | 258 |

Issue number | 3 |

DOIs | |

Publication status | Published - Feb 1 2010 |

### Keywords

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

### ASJC Scopus subject areas

- Analysis