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Local continuity modulus for wiener and infinite dimensional OU processes

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Abstract

In this paper, we gave a proof for the local continuity modulus theorem of the Wiener process, i.e.,

$$\mathop {\lim }\limits_{t \to 0} \mathop {\sup }\limits_{0 \leqslant s \leqslant t} |W(s)|/(2s\log \log (1/s))^{1/2} = 1$$

a.s. This result was given by Csörgö and Révész (1981), but the proof gets them nowhere. We also gave a similar local continuity modulus result for the infinite dimensional OU processes.

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References

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Authors and Affiliations

  1. Department of Mathematics, Hangzhou University, 310028, Hangzhou, China

    Lu Chuanrong & Shen Siwei

Authors
  1. Lu Chuanrong
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  2. Shen Siwei
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  3. Wang Xiuyun
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Supported by the National Natural Science Fund and Zhejing Provincial Natural Science Foundation.

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Chuanrong, L., Siwei, S. & Xiuyun, W. Local continuity modulus for wiener and infinite dimensional OU processes. Acta Mathematica Sinica 10, 219–224 (1994). https://doi.org/10.1007/BF02580430

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  • DOI: https://doi.org/10.1007/BF02580430

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