### Abstract

Malliavin calculus is applicable to functionals of stable processes by using subordination. We prepare Malliavin calculus for stochastic differential equations driven by Brownian motions with deterministic time change, and the conditions that the existence and the regularity of the densities inherit from those of the densities of conditional probabilities. By using these, we prove regularity properties of the solutions of equations driven by subordinated Brownian motions.[4] a similar problem is considered.this article we consider more general cases. We also consider equations driven by rotation-invariant stable processes. We prove that the ellipticity of the equations implies the existence of the density of the solution, and we also prove that the regularity of the coefficients implies the regularity of the densities in the case when the equations are driven by one rotation-invariant stable process.

Original language | English |
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Pages (from-to) | 491-520 |

Number of pages | 30 |

Journal | Kyoto Journal of Mathematics |

Volume | 50 |

Issue number | 3 |

DOIs | |

Publication status | Published - Sep 2010 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Malliavin calculus for stochastic differential equations driven by subordinated Brownian motions.** / Kusuoka, Seiichiro.

Research output: Contribution to journal › Article

*Kyoto Journal of Mathematics*, vol. 50, no. 3, pp. 491-520. https://doi.org/10.1215/0023608X-2010-003

}

TY - JOUR

T1 - Malliavin calculus for stochastic differential equations driven by subordinated Brownian motions

AU - Kusuoka, Seiichiro

PY - 2010/9

Y1 - 2010/9

N2 - Malliavin calculus is applicable to functionals of stable processes by using subordination. We prepare Malliavin calculus for stochastic differential equations driven by Brownian motions with deterministic time change, and the conditions that the existence and the regularity of the densities inherit from those of the densities of conditional probabilities. By using these, we prove regularity properties of the solutions of equations driven by subordinated Brownian motions.[4] a similar problem is considered.this article we consider more general cases. We also consider equations driven by rotation-invariant stable processes. We prove that the ellipticity of the equations implies the existence of the density of the solution, and we also prove that the regularity of the coefficients implies the regularity of the densities in the case when the equations are driven by one rotation-invariant stable process.

AB - Malliavin calculus is applicable to functionals of stable processes by using subordination. We prepare Malliavin calculus for stochastic differential equations driven by Brownian motions with deterministic time change, and the conditions that the existence and the regularity of the densities inherit from those of the densities of conditional probabilities. By using these, we prove regularity properties of the solutions of equations driven by subordinated Brownian motions.[4] a similar problem is considered.this article we consider more general cases. We also consider equations driven by rotation-invariant stable processes. We prove that the ellipticity of the equations implies the existence of the density of the solution, and we also prove that the regularity of the coefficients implies the regularity of the densities in the case when the equations are driven by one rotation-invariant stable process.

UR - http://www.scopus.com/inward/record.url?scp=79957803737&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79957803737&partnerID=8YFLogxK

U2 - 10.1215/0023608X-2010-003

DO - 10.1215/0023608X-2010-003

M3 - Article

AN - SCOPUS:79957803737

VL - 50

SP - 491

EP - 520

JO - Journal of Mathematics of Kyoto University

JF - Journal of Mathematics of Kyoto University

SN - 0023-608X

IS - 3

ER -