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On bifractional Brownian motion. (English) Zbl 1100.60019

The bifractional Brownian motion \((B_t^{H,K})\) is defined, for \(0< H<1\) and \(0<K\leq 1\), as the centered Gaussian process started from 0 having covariance: \[ (s,t)\mapsto 2^{-K}[(s^{2H}+t^{2H})^K-|s-t|^{2HK}]. \] It is \(HK\)-self-similar, and Hölder continuous of any order \(<HK\). The authors establish in particular that:
– \(\lim_{\varepsilon \searrow 0}\int^t_0|B_{s+\varepsilon}^{H,K}-B_s^{H,K}|^{1/HK}ds= c_{H,K}\times t\) exists in probability;
– \(B^{H,K}\) is not a semi-martingale, nor a Markov process (of course except if \(2H=K=1)\);
– \((B^{H,K}+W)\) has the law of \(W\), if \(HK>\frac 34\), for any independent Wiener process \(W\);
– \(B^{H,K}\) has short memory if \(2HK\leq 1\);
– \(B^{H,K}\) admits a local time, in the space \(\mathbb{D}^{\alpha,2}\) for \(\alpha<\frac {1}{2HK}-\frac 12\).

MSC:

60G18 Self-similar stochastic processes
60G15 Gaussian processes
60H05 Stochastic integrals
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References:

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