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Sample Path Properties of Bifractional Brownian Motion

# Sample Path Properties of Bifractional Brownian Motion,Ciprian A. Tudor,Yimin Xiao

Sample Path Properties of Bifractional Brownian Motion
Let $B^{H, K}= \big\{B^{H, K}(t), t \in \R_+ \big\}$ be a bifractional Brownian motion in $\R^d$. We prove that $B^{H, K}$ is strongly locally nondeterministic. Applying this property and a stochastic integral representation of $B^{H, K}$, we establish Chung's law of the iterated logarithm for $B^{H, K}$, as well as sharp H\"older conditions and tail probability estimates for the local times of $B^{H, K}$. We also consider the existence and the regularity of the local times of multiparameter bifractional Brownian motion $B^{\bar{H}, \bar{K}}= \big\{B^{\bar{H}, \bar{K}}(t), t \in \R^N_+ \big\}$ in $\R^d$ using Wiener-It\^o chaos expansion.
Published in 2006.
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## Citation Context (6)

• ...A stochastic analysis for this process can be found in [7] and a study of its occupation density has been done in [2] and [14]...

### Khalifa Es-Sebaiy, et al. Occupation densities for certain processes related to fractional Brown...

• ...so relation (2.5) holds with � = HK. A stochastic analysis for this process can be found in [7] and a study of its occupation densities has been done in [2], [12]...

### Khalifa Es-Sebaiy, et al. Occupation densities for certain processes related to fractional Brown...

• ...Introduced in [4], the bifractional Brownian motion, a generalization of the fractional Brownian motion, has been studied in many aspects (see [1], [3], [6], [7], [8], [9] and [10])...

### Makoto Maejima, et al. Limits of bifractional Brownian noises

• ...The process bfBm is also studied in [14, 19]...

### Tomasz Bojdecki, et al. Some extensions of fractional Brownian motion and sub-fractional Brown...

• ...Other papers treated different aspects of this stochastic process, like sample pats properties, extension of the parameters or statistical applications (see [6], [4], [22] or [11]...
• ...We start with the one dimensional bifBm and we first derive an Itˆo and an Tanaka formula for it when 2HK ≥ 1. We mention that the Itˆo formula has been already proved by [15] but here we propose an alternative proof based on the Taylor expansion which appears to be also useful in the multidimensional settings...
• ...It follows actually from [15] that the space |H| is a Banach space for the norm k � k |H| and it is included in H. In fact,...
• ...This paragraph is consecrated to the proof of Itˆo formula and Tanaka formula for the onedimensional bifractional Brownian motion with 2HK ≥ 1. Note that the Itˆo formula has been already proved in [15]; here we propose a different approach based on the Taylor expansion which be also used in the multidimensional settings...
• ...This term (in fact, slightly modified) appeared in some other papers such as Proposition 12 in [10], or in [22]...

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