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 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.
In: Journal of Functional Analysis, Vol. 258, No. 3, 01.02.2010, p. 758-784.Research output: Contribution to journal › Article
}
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 -