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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2025-01-01
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author | Hyun-Suk Park |
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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 |