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Density Bounds for Solutions to Differential Equations Driven by Gaussian Rough Paths

Author

Listed:
  • Benjamin Gess

    (Max Planck Institute for Mathematics in the Sciences
    University of Bielefeld)

  • Cheng Ouyang

    (University of Illinois at Chicago)

  • Samy Tindel

    (Purdue University)

Abstract

We consider finite-dimensional rough differential equations driven by centered Gaussian processes. Combining Malliavin calculus, rough paths techniques and interpolation inequalities, we establish upper bounds on the density of the corresponding solution for any fixed time $$t>0$$t>0. In addition, we provide Varadhan estimates for the asymptotic behavior of the density for small noise. The emphasis is on working with general Gaussian processes with covariance function satisfying suitable abstract, checkable conditions.

Suggested Citation

  • Benjamin Gess & Cheng Ouyang & Samy Tindel, 2020. "Density Bounds for Solutions to Differential Equations Driven by Gaussian Rough Paths," Journal of Theoretical Probability, Springer, vol. 33(2), pages 611-648, June.
    • Handle: RePEc:spr:jotpro:v:33:y:2020:i:2:d:10.1007_s10959-019-00967-0
    DOI: 10.1007/s10959-019-00967-0
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    References listed on IDEAS

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    1. Lou, Shuwen & Ouyang, Cheng, 2017. "Local times of stochastic differential equations driven by fractional Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 127(11), pages 3643-3660.
    2. Baudoin, Fabrice & Ouyang, Cheng, 2011. "Small-time kernel expansion for solutions of stochastic differential equations driven by fractional Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 121(4), pages 759-792, April.
    3. Russo, Francesco & Tudor, Ciprian A., 2006. "On bifractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 116(5), pages 830-856, May.
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    Cited by:

    1. Yuzuru Inahama & Bin Pei, 2022. "Positivity of the Density for Rough Differential Equations," Journal of Theoretical Probability, Springer, vol. 35(3), pages 1863-1877, September.
    2. Song, Jian & Tindel, Samy, 2022. "Skorohod and Stratonovich integrals for controlled processes," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 569-595.

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