Spherical and Hyperbolic Fractional Brownian Motion

By J. Istas. 2005

62 citations recorded by the source corpus.

We define a Fractional Brownian Motion indexed by a sphere, or more generally by a compact rank one symmetric space, and prove that it exists if, and only if, $0 < H \leq 1/2$. We then prove that Fractional Brownian Motion indexed by an hyperbolic space exists if, and only if, $0 < H \leq 1/2$. At last, we prove that Fractional Brownian Motion indexed by a real tree exists when $0 < H \leq 1/2$.

Subfields: Economics and Econometrics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7859c0014f6616db05ee965c494a6dfe3dfb6aab

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In Economics and Econometrics

1. Jean Picard
2. Yimin Xiao
3. Dan Cheng

In Finance

1. Jean Picard
2. Yimin Xiao
3. Dan Cheng
4. Dongsheng Wu
5. Davar Khoshnevisan
6. Antoine Ayache
7. Hermine Bierm'e
8. C. Lacaux

In Mathematical Physics

1. Jean Picard
2. Yimin Xiao
3. Dan Cheng
4. Dongsheng Wu
5. Antoine Ayache
6. Davar Khoshnevisan

Citations in the corpus

• Extremes of independent Gaussian processes (2009, unknown): “[6], that the kernel Γ(t1, t2) := ρ(t1, t2) is negative definite for β ∈ (0, 1).”
• Representation formulae for the fractional Brownian motion (2009, methodology): “This result was proved in [18] when H ≤ 1/2, and the case H > 1/2 was studied in [17] with an approximation method.”
• Excursion probability of Gaussian random fields on sphere (2014, background): “The spherical fractional Brownian motion (SFBM) Bβ = {Bβ(x): x ∈ SN} is defined by Istas [17] as a centered real-valued Gaussian random field such that Bβ(o) = 0 and E(Bβ(x)−Bβ(y))(2) = d(x, y) ∀x, y ∈ S , where β ∈ (0,1/2].”