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Smoothness for the collision local times of bifractional Brownian motions. (English) Zbl 1236.60042

Summary: Let \(B^{H_i ,K_i } = \{ B_t^{H_i ,K_i } , \, t \geqslant 0\}\), \(i = 1, 2\) be two independent bifractional Brownian motions with respective indices \(H _{i } \in (0, 1)\) and \(K _{i } \in (0, 1]\). One of the main motivations of this paper is to investigate the smoothness of the collision local time, introduced by Y. Jiang and Y. Wang [Sci. China, Ser. A 52, No. 9, 1905–1919 (2009; Zbl 1181.60059)], \[ \ell _T = \int_0^T \delta (B_s^{H_1 ,K_1 } - B_s^{H_2 ,K_2 })\,ds,\;T > 0, \] where \(\delta \) denotes the Dirac delta function. By an elementary method, we show that \(\ell _{T }\) is smooth in the sense of Meyer-Watanabe if and only if \(\min\{H _{1} K _{1}, H _{2} K _{2}\} < 1/3\).

MSC:

60G22 Fractional processes, including fractional Brownian motion
60G15 Gaussian processes
60H07 Stochastic calculus of variations and the Malliavin calculus
60J55 Local time and additive functionals

Citations:

Zbl 1181.60059
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References:

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