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.
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 -