Density Formula in Malliavin Calculus by Using Stein’s Method and Diffusions

Let <i>G</i> be a random variable of functionals of an isonormal Gaussian process <i>X</i> defined on some probability space. Studies have been conducted to determine the exact form of the density function of the random variable <i>G</i>. In this paper, unlike pre...

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Main Author: Hyun-Suk Park
Format: Article
Language:English
Published: MDPI AG 2025-01-01
Series:Mathematics
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Online Access:https://www.mdpi.com/2227-7390/13/2/323
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author Hyun-Suk Park
author_facet Hyun-Suk Park
author_sort Hyun-Suk Park
collection DOAJ
description Let <i>G</i> be a random variable of functionals of an isonormal Gaussian process <i>X</i> defined on some probability space. Studies have been conducted to determine the exact form of the density function of the random variable <i>G</i>. In this paper, unlike previous studies, we will use the Stein’s method for invariant measures of diffusions to obtain the density formula of <i>G</i>. By comparing the density function obtained in this paper with that of the diffusion invariant measure, we find that the diffusion coefficient of an Itô diffusion with an invariant measure having a density can be expressed as in terms of operators in Malliavin calculus.
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spelling doaj-art-206a39d90c8648988766fe56977f19b72025-01-24T13:40:11ZengMDPI AGMathematics2227-73902025-01-0113232310.3390/math13020323Density Formula in Malliavin Calculus by Using Stein’s Method and DiffusionsHyun-Suk Park0Division of Data Science, Data Science Convergence Research Center, Hallym University, Chuncheon 24252, Republic of KoreaLet <i>G</i> be a random variable of functionals of an isonormal Gaussian process <i>X</i> defined on some probability space. Studies have been conducted to determine the exact form of the density function of the random variable <i>G</i>. In this paper, unlike previous studies, we will use the Stein’s method for invariant measures of diffusions to obtain the density formula of <i>G</i>. By comparing the density function obtained in this paper with that of the diffusion invariant measure, we find that the diffusion coefficient of an Itô diffusion with an invariant measure having a density can be expressed as in terms of operators in Malliavin calculus.https://www.mdpi.com/2227-7390/13/2/323Malliavin calculusStein’s methoddensity functionstandard normal random variableItô diffusion
spellingShingle Hyun-Suk Park
Density Formula in Malliavin Calculus by Using Stein’s Method and Diffusions
Mathematics
Malliavin calculus
Stein’s method
density function
standard normal random variable
Itô diffusion
title Density Formula in Malliavin Calculus by Using Stein’s Method and Diffusions
title_full Density Formula in Malliavin Calculus by Using Stein’s Method and Diffusions
title_fullStr Density Formula in Malliavin Calculus by Using Stein’s Method and Diffusions
title_full_unstemmed Density Formula in Malliavin Calculus by Using Stein’s Method and Diffusions
title_short Density Formula in Malliavin Calculus by Using Stein’s Method and Diffusions
title_sort density formula in malliavin calculus by using stein s method and diffusions
topic Malliavin calculus
Stein’s method
density function
standard normal random variable
Itô diffusion
url https://www.mdpi.com/2227-7390/13/2/323
work_keys_str_mv AT hyunsukpark densityformulainmalliavincalculusbyusingsteinsmethodanddiffusions