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

Research output: Contribution to journalArticle

5 Citations (Scopus)

### 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 2499-2506 8 Statistics and Probability Letters 83 11 https://doi.org/10.1016/j.spl.2013.07.011 Published - 2013 Nov Yes

### Fingerprint

Continuity
Stochastic differential equations
Coefficients

### Keywords

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

### ASJC Scopus subject areas

• Statistics, Probability and Uncertainty
• Statistics and Probability

### Cite this

In: Statistics and Probability Letters, Vol. 83, No. 11, 11.2013, p. 2499-2506.

Research output: Contribution to journalArticle

@article{064df4e65d944ecab3fdcb6f3d05ae9d,
title = "Absolute continuity of the laws of a multi-dimensional stochastic differential equation with coefficients dependent on the maximum",
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.",
keywords = "60H07, 60H10, Absolutely continuous law, Malliavin calculus, Stochastic differential equation",
author = "Tomonori Nakatsu",
year = "2013",
month = "11",
doi = "10.1016/j.spl.2013.07.011",
language = "English",
volume = "83",
pages = "2499--2506",
journal = "Statistics and Probability Letters",
issn = "0167-7152",
publisher = "Elsevier",
number = "11",

}

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

AN - SCOPUS:84882769620

VL - 83

SP - 2499

EP - 2506

JO - Statistics and Probability Letters

JF - Statistics and Probability Letters

SN - 0167-7152

IS - 11

ER -