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

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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 \(t>0\). 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.

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Notes

  1. As pointed out, for example, in [23] this process does not fit in the Volterra framework.

  2. We may ignore the (constant, random) zero-mode in the series since we are only interested in properties of the increments of the process.

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Acknowledgements

We would like to thank the anonymous referee for his/her very careful reading of the first version of this paper and for many valuable suggestions.

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Authors and Affiliations

  1. Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany

    Benjamin Gess

  2. University of Bielefeld, Bielefeld, Germany

    Benjamin Gess

  3. Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 S. Morgan St., Chicago, IL, 60607, USA

    Cheng Ouyang

  4. Department of Mathematics, Purdue University, 150 N. University St., West Lafayette, IN, 47907-2067, USA

    Samy Tindel

Authors
  1. Benjamin Gess
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  2. Cheng Ouyang
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  3. Samy Tindel
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Correspondence to Cheng Ouyang.

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C. Ouyang’ research is supported in part by Simons Grant #355480. S. Tindel is supported in part by NSF Grant DMS 0907326.

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Gess, B., Ouyang, C. & Tindel, S. Density Bounds for Solutions to Differential Equations Driven by Gaussian Rough Paths. J Theor Probab 33, 611–648 (2020). https://doi.org/10.1007/s10959-019-00967-0

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  • DOI: https://doi.org/10.1007/s10959-019-00967-0

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