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On the collision local time of bifractional Brownian motions. (English) Zbl 1180.60034

Summary: Let \(B^{H_i,K_i}= \{B_t^{H_i,K_i}\), \(t\geq 0\}\), \(i=1,2\) be two independent bifractional Brownian motions of dimension 1, with indices \(H_i\in(0,1)\) and \(K_i\in(0,1]\). We investigate the collision local time of bifractional Brownian motions
\[ \ell_T= \int_0^T \delta(B_t^{H_1,K_1}- B_t^{H_2,K_2}),\,dt, \quad 0<T<\infty \]
where \(\delta\) denotes the Dirac delta function at zero. We show that \(\ell_T\) exists in \(L^2\), and it is Hölder continuous of order \(1-\min\{H_1K_1,H_2K_2\}\), and furthermore, it is also smooth in the sense of Meyer-Watanabe.

MSC:

60G22 Fractional processes, including fractional Brownian motion
60G15 Gaussian processes
60G18 Self-similar stochastic processes
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