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Smoothness for the collision local time of two multidimensional bifractional Brownian motions

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Abstract

Let \({B^{{H_i},{K_i}}} = \{ B_t^{{H_i},{K_i}},t \ge 0\} ,{\rm{ }}i = 1,2\) be two independent, d-dimensional bifractional Brownian motions with respective indices H i ∈ (0, 1) and K i ∈ (0, 1]. Assume d ⩾ 2. One of the main motivations of this paper is to investigate smoothness of the collision local time

$${l_T} = \int_0^T {\delta (B_s^{{H_1},{K_1}} - B_s^{{H_2},{K_2}}} )ds,{\rm{ }}T > 0,$$

, where δ denotes the Dirac delta function. By an elementary method we show that l T is smooth in the sense of Meyer-Watanabe if and only if min{H 1 K 1,H 2 K 2} < 1/(d + 2).

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Authors and Affiliations

  1. Department of Mathematics, Anhui Normal University, 1 East Beijing Rd., Wuhu, 241000, P.R. China

    Guangjun Shen

  2. Department of Mathematics, Donghua University, 2999 North Renmin Rd., Songjiang, Shanghai, 201620, P.R. China

    Litan Yan

  3. Department of Mathematics, East China University of Science and Technology, 130 Mei Long Rd., Xuhui, Shanghai, 200237, P.R. China

    Chao Chen

Authors
  1. Guangjun Shen
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  2. Litan Yan
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  3. Chao Chen
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Correspondence to Guangjun Shen.

Additional information

Guangjun Shen is partially supported by National Natural Science Foundation of China (11271020), Natural Science Foundation of Anhui Province (1208085MA11) and Key Natural Science Foundation of Anhui Educational Committee (KJ2011A139), Litan Yan is partially supported by National Natural Science Foundation of China (11171062), Innovation Program of Shanghai Municipal Education Commission (12ZZ063).

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Shen, G., Yan, L. & Chen, C. Smoothness for the collision local time of two multidimensional bifractional Brownian motions. Czech Math J 62, 969–989 (2012). https://doi.org/10.1007/s10587-012-0077-7

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  • DOI: https://doi.org/10.1007/s10587-012-0077-7

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