Joint continuity of the local times of fractional Brownian sheets

By Antoine Ayache, Dongsheng Wu, Yimin Xiao. 2008

62 citations recorded by the source corpus.

Let $B^H=\{B^H(t),t\in{{\mathbb{R}}_+^N}\}$ be an $(N,d)$-fractional Brownian sheet with index $H=(H_1,...,H_N)\in(0,1)^N$ defined by $B^H(t)=(B^H_1(t),...,B^H_d(t)) (t\in {\mathbb{R}}_+^N),$ where $B^H_1,...,B^H_d$ are independent copies of a real-valued fractional Brownian sheet $B_0^H$. We prove that if $d<\sum_{\ell=1}^NH_{\ell}^{-1}$, then the local times of $B^H$ are jointly continuous. This verifies a conjecture of Xiao and Zhang (Probab. Theory Related Fields 124 (2002)). We also establish sharp local and global H\"{o}lder conditions for the local times of $B^H$. These results are applied to study analytic and geometric properties of the sample paths of $B^H$.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7f885dc34f4a3e3f41fbb3b68611055dedcd81a1

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Finance

1. Dongsheng Wu
2. Yimin Xiao
3. Davar Khoshnevisan
4. Antoine Ayache
5. Hermine Bierm'e
6. C. Lacaux

In Mathematical Physics

1. Dongsheng Wu
2. Yimin Xiao
3. Antoine Ayache
4. Davar Khoshnevisan

Citations in the corpus

• Sample Path Properties of Anisotropic Gaussian Random Fields (2009, background): “Ayache, Wu and Xiao [ 6 ] established the joint continuity of the local times under the optimal condition and studied the local and uniform H¨ older conditions for the maximum local times.”
• Geometric Properties of Fractional Brownian Sheets (2007, background): “The property of sectorial local nondeterminism not only plays an important role in this paper, but also in studying other problems such as local times of fractional Brownian sheets [see Ayache, Wu and Xiao [4]].”