### Abstract

In this article, we consider an m-dimensional stochastic differential equation with coefficients which depend on the maximum of the solution. First, we prove the absolute continuity of the law of the solution. Then we prove that the joint law of the maximum of the ith component of the solution and the ^{i '}th component of the solution is absolutely continuous with respect to the Lebesgue measure in a particular case. The main tool to prove the absolute continuity of the laws is Malliavin calculus.

Original language | English |
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Pages (from-to) | 2499-2506 |

Number of pages | 8 |

Journal | Statistics and Probability Letters |

Volume | 83 |

Issue number | 11 |

DOIs | |

Publication status | Published - 2013 Nov |

Externally published | Yes |

### Fingerprint

### Keywords

- 60H07
- 60H10
- Absolutely continuous law
- Malliavin calculus
- Stochastic differential equation

### ASJC Scopus subject areas

- Statistics, Probability and Uncertainty
- Statistics and Probability

### Cite this

**Absolute continuity of the laws of a multi-dimensional stochastic differential equation with coefficients dependent on the maximum.** / Nakatsu, Tomonori.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Absolute continuity of the laws of a multi-dimensional stochastic differential equation with coefficients dependent on the maximum

AU - Nakatsu, Tomonori

PY - 2013/11

Y1 - 2013/11

N2 - In this article, we consider an m-dimensional stochastic differential equation with coefficients which depend on the maximum of the solution. First, we prove the absolute continuity of the law of the solution. Then we prove that the joint law of the maximum of the ith component of the solution and the i 'th component of the solution is absolutely continuous with respect to the Lebesgue measure in a particular case. The main tool to prove the absolute continuity of the laws is Malliavin calculus.

AB - In this article, we consider an m-dimensional stochastic differential equation with coefficients which depend on the maximum of the solution. First, we prove the absolute continuity of the law of the solution. Then we prove that the joint law of the maximum of the ith component of the solution and the i 'th component of the solution is absolutely continuous with respect to the Lebesgue measure in a particular case. The main tool to prove the absolute continuity of the laws is Malliavin calculus.

KW - 60H07

KW - 60H10

KW - Absolutely continuous law

KW - Malliavin calculus

KW - Stochastic differential equation

UR - http://www.scopus.com/inward/record.url?scp=84882769620&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84882769620&partnerID=8YFLogxK

U2 - 10.1016/j.spl.2013.07.011

DO - 10.1016/j.spl.2013.07.011

M3 - Article

VL - 83

SP - 2499

EP - 2506

JO - Statistics and Probability Letters

JF - Statistics and Probability Letters

SN - 0167-7152

IS - 11

ER -