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Bound for an Approximation of Invariant Density of Diffusions via Density Formula in Malliavin Calculus

Author

Listed:
  • Yoon-Tae Kim

    (Division of Data Science and Data Science Convergence Research Center, Hallym University, Chuncheon 24252, Republic of Korea)

  • Hyun-Suk Park

    (Division of Data Science and Data Science Convergence Research Center, Hallym University, Chuncheon 24252, Republic of Korea)

Abstract

The Kolmogorov and total variation distance between the laws of random variables have upper bounds represented by the L 1 -norm of densities when random variables have densities. In this paper, we derive an upper bound, in terms of densities such as the Kolmogorov and total variation distance, for several probabilistic distances (e.g., Kolmogorov distance, total variation distance, Wasserstein distance, Forter–Mourier distance, etc.) between the laws of F and G in the case where a random variable F follows the invariant measure that admits a density and a differentiable random variable G , in the sense of Malliavin calculus, and also allows a density function.

Suggested Citation

  • Yoon-Tae Kim & Hyun-Suk Park, 2023. "Bound for an Approximation of Invariant Density of Diffusions via Density Formula in Malliavin Calculus," Mathematics, MDPI, vol. 11(10), pages 1-18, May.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:10:p:2302-:d:1147453
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    References listed on IDEAS

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    1. Kim, Yoon Tae & Park, Hyun Suk, 2018. "An Edgeworth expansion for functionals of Gaussian fields and its applications," Stochastic Processes and their Applications, Elsevier, vol. 128(12), pages 3967-3999.
    2. Kusuoka, Seiichiro & Tudor, Ciprian A., 2012. "Stein’s method for invariant measures of diffusions via Malliavin calculus," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1627-1651.
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    Cited by:

    1. Alexander N. Tikhomirov & Vladimir V. Ulyanov, 2023. "On the Special Issue “Limit Theorems of Probability Theory”," Mathematics, MDPI, vol. 11(17), pages 1-4, August.

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