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Besov-Orlicz Path Regularity of Non-Gaussian Processes

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Abstract

In the article, Besov-Orlicz regularity of sample paths of stochastic processes that are represented by multiple integrals of order \(n\in \mathbb {N}\) is treated. We assume that the considered processes belong to the Hölder space

$$ C^{\alpha}([0,T];L^{2}({\Omega}))\quad\text{with}\quad \alpha\in (0,1), $$

and we give sufficient conditions for them to have paths in the exponential Besov-Orlicz space

$$ B_{{\varPhi}_{2/n},\infty}^{\alpha}(0,T)\qquad \text{with}\qquad {\varPhi}_{2/n}(x)=\mathrm{e}^{x^{2/n}}-1. $$

These results provide an extension of what is known for scalar Gaussian stochastic processes to stochastic processes in an arbitrary finite Wiener chaos. As an application, the Besov-Orlicz path regularity of fractionally filtered Hermite processes is studied. But while the main focus is on the non-Gaussian case, some new path properties are obtained even for fractional Brownian motions.

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Acknowledgements

The authors are grateful to the anonymous reviewers for their careful reading of the article and for their helpful remarks and suggestions.

Funding

This research was supported by the Czech Science Foundation project No. 19-07140S.

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Authors and Affiliations

  1. Charles University, Faculty of Mathematics and Physics, Sokolovská 83, 186 75, Prague 8, Czech Republic

    Petr Čoupek

  2. Czech Academy of Sciences, Institute of Information Theory and Automation, Pod Vodárenskou věží 4, 182 00, Prague 8, Czech Republic

    Martin Ondreját

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  1. Petr Čoupek
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Correspondence to Martin Ondreját.

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Čoupek, P., Ondreját, M. Besov-Orlicz Path Regularity of Non-Gaussian Processes. Potential Anal 60, 307–339 (2024). https://doi.org/10.1007/s11118-022-10051-8

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