We consider the existence and Hölder continuity conditions for the k-th-order derivatives of self-intersection local time for d-dimensional fractional Brownian motion, where \(k=(k_1,k_2,\ldots , k_d)\). Moreover, we show a limit theorem for the critical case with \(H=\frac{2}{3}\) and \(d=1\), which was conjectured by Jung and Markowsky [7].

中文翻译：

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分数布朗运动的自交局部时间的高阶导数

我们考虑d维分数布朗运动的自交局部时间的k阶导数的存在性和Hölder连续性条件，其中\（k =（k_1，k_2，\ ldots，k_d）\）。此外，我们用\（H = \ frac {2} {3} \）和\（d = 1 \）给出了临界情况的极限定理，这是由Jung和Markowsky提出的[7]。