### 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 |
---|---|

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 |

Externally published | Yes |

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### 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.

Research output: Contribution to journal › Article

*Journal of Functional Analysis*, vol. 258, no. 3, pp. 758-784. https://doi.org/10.1016/j.jfa.2009.09.009

}

TY - JOUR

T1 - Existence of densities of solutions of stochastic differential equations by Malliavin calculus

AU - Kusuoka, Seiichiro

PY - 2010/2/1

Y1 - 2010/2/1

N2 - 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.

AB - 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.

KW - Absolute continuity

KW - Existence of densities

KW - Existence of fundamental solutions

KW - Malliavin calculus

KW - Stochastic differential equation

UR - http://www.scopus.com/inward/record.url?scp=70350588988&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70350588988&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2009.09.009

DO - 10.1016/j.jfa.2009.09.009

M3 - Article

AN - SCOPUS:70350588988

VL - 258

SP - 758

EP - 784

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 3

ER -