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On the collision local time of sub-fractional Brownian motions. (English) Zbl 1185.60040
Summary: Let $S^{H_i} = \{S^{H_i}_t,t\geq 0\}, i=1,2,$ be two independent sub-fractional Brownian motions with respective indices $H_i \in (0,1)$. We consider the so-called collision local time $$\ell _T = \int _0^T \delta (S_t^{H_1}-S_t^{H_2})\text dt,\quad T>0,$$ where $\delta $ denotes the Dirac delta function. By an elementary method we show that $\ell _T$ is smooth in the sense of Meyer and Watanabe if and only if min$\{H_1,H_2\}< 1/3.$

60G22Fractional processes, including fractional Brownian motion
60G15Gaussian processes
60G18Self-similar processes
60F25$L^p$-limit theorems (probability)
Full Text: DOI
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