TY - JOUR

T1 - Some Properties of Density Functions on Maxima of Solutions to One-Dimensional Stochastic Differential Equations

AU - Nakatsu, Tomonori

N1 - Funding Information:
The author would like to thank Masafumi Hayashi and Arturo Kohatsu-Higa for fruitful discussions on the topics related to this article. The author is grateful to the anonymous reviewer for careful reading of the manuscript and helpful suggestions. This work was supported by JSPS KAKENHI Grant Number 17K14209.
Publisher Copyright:
© 2019, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2019/12/1

Y1 - 2019/12/1

N2 - This article proves some properties of the probability density function concerning maxima of a solution to one-dimensional stochastic differential equations. We first obtain lower and upper bounds on the density function of the discrete time maximum of the solution. We then prove that the density function of the discrete time maximum converges to that of the continuous time maximum of the solution. Finally, we prove the positivity of the density function of the continuous time maximum and a relationship between the density functions of the continuous time maximum and the solution itself.

AB - This article proves some properties of the probability density function concerning maxima of a solution to one-dimensional stochastic differential equations. We first obtain lower and upper bounds on the density function of the discrete time maximum of the solution. We then prove that the density function of the discrete time maximum converges to that of the continuous time maximum of the solution. Finally, we prove the positivity of the density function of the continuous time maximum and a relationship between the density functions of the continuous time maximum and the solution itself.

KW - Continuous/discrete time maximum

KW - Malliavin calculus

KW - Probability density function

KW - Stochastic differential equation

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U2 - 10.1007/s10959-019-00885-1

DO - 10.1007/s10959-019-00885-1

M3 - Article

AN - SCOPUS:85061805762

VL - 32

SP - 1746

EP - 1779

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

SN - 0894-9840

IS - 4

ER -