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Stein’s method for invariant measures of diffusions via Malliavin calculus

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  • Kusuoka, Seiichiro
  • Tudor, Ciprian A.

Abstract

Given a random variable F regular enough in the sense of the Malliavin calculus, we are able to measure the distance between its law and any probability measure with a density function which is continuous, bounded, strictly positive on an interval in the real line and admits finite variance. The bounds are given in terms of the Malliavin derivative of F. Our approach is based on the theory of Itô diffusions and the stochastic calculus of variations. Several examples are considered in order to illustrate our general results.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:122:y:2012:i:4:p:1627-1651
    DOI: 10.1016/j.spa.2012.02.005
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    References listed on IDEAS

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    1. Viens, Frederi G., 2009. "Stein's lemma, Malliavin calculus, and tail bounds, with application to polymer fluctuation exponent," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3671-3698, October.
    2. Ait-Sahalia, Yacine, 1996. "Nonparametric Pricing of Interest Rate Derivative Securities," Econometrica, Econometric Society, vol. 64(3), pages 527-560, May.
    3. Bo Martin Bibby & Michael Sørensen, 2001. "Simplified Estimating Functions for Diffusion Models with a High‐dimensional Parameter," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 28(1), pages 99-112, March.
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    1. Gaunt, Robert E., 2019. "Stein operators for variables form the third and fourth Wiener chaoses," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 118-126.
    2. Peng Chen & Ivan Nourdin & Lihu Xu & Xiaochuan Yang & Rui Zhang, 2022. "Non-integrable Stable Approximation by Stein’s Method," Journal of Theoretical Probability, Springer, vol. 35(2), pages 1137-1186, June.
    3. Ley, Christophe, 2023. "When the score function is the identity function - A tale of characterizations of the normal distribution," Econometrics and Statistics, Elsevier, vol. 26(C), pages 153-160.
    4. Arras, Benjamin & Azmoodeh, Ehsan & Poly, Guillaume & Swan, Yvik, 2019. "A bound on the Wasserstein-2 distance between linear combinations of independent random variables," Stochastic Processes and their Applications, Elsevier, vol. 129(7), pages 2341-2375.
    5. Christophe Ley & Gesine Reinert & Yvik Swan, 2014. "Approximate Computation of Expectations: the Canonical Stein Operator," Working Papers ECARES ECARES 2014-36, ULB -- Universite Libre de Bruxelles.
    6. Eden, Richard & Víquez, Juan, 2015. "Nourdin–Peccati analysis on Wiener and Wiener–Poisson space for general distributions," Stochastic Processes and their Applications, Elsevier, vol. 125(1), pages 182-216.
    7. Tudor, Ciprian A., 2014. "Chaos expansion and asymptotic behavior of the Pareto distribution," Statistics & Probability Letters, Elsevier, vol. 91(C), pages 62-68.
    8. 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.
    9. Hyun-Suk Park, 2025. "Density Formula in Malliavin Calculus by Using Stein’s Method and Diffusions," Mathematics, MDPI, vol. 13(2), pages 1-15, January.
    10. Privault, N. & Yam, S.C.P. & Zhang, Z., 2019. "Poisson discretizations of Wiener functionals and Malliavin operators with Wasserstein estimates," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3376-3405.

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