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Hausdorff and Fourier dimension of graph of continuous additive processes

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  • Dysthe, Dexter
  • Lai, Chun-Kit

Abstract

An additive process is a stochastic process with independent increments and that is continuous in probability. In this paper, we study the almost sure Hausdorff and Fourier dimension of the graph of continuous additive processes with zero mean. Such processes can be represented as Xt=BV(t) where B is Brownian motion and V is a continuous increasing function. We show that these dimensions depend on the local uniform Hölder indices. In particular, if V is locally uniformly bi-Lipschitz, then the Hausdorff dimension of the graph will be 3/2. We also show that the Fourier dimension almost surely is positive if V admits at least one point with positive lower Hölder regularity.

Suggested Citation

  • Dysthe, Dexter & Lai, Chun-Kit, 2023. "Hausdorff and Fourier dimension of graph of continuous additive processes," Stochastic Processes and their Applications, Elsevier, vol. 155(C), pages 355-392.
    • Handle: RePEc:eee:spapps:v:155:y:2023:i:c:p:355-392
    DOI: 10.1016/j.spa.2022.10.010
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    References listed on IDEAS

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    1. Yang, Ming, 2008. "Hausdorff dimension of the image of additive processes," Stochastic Processes and their Applications, Elsevier, vol. 118(4), pages 681-702, April.
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