REGULARITY - 2011 - Annual activity report

REGULARITY

REGULARITY - 2011

Team Regularity

Members

Overall Objectives

Scientific Foundations

Application Domains

Software

FracLab

New Results

Contracts and Grants with Industry

Grants with Industry

Partnerships and Cooperations

Dissemination

Bibliography

Previous |

Home | Next next

Section: New Results

Hölder regularity of Set-Indexed processes

Participants : Erick Herbin, Alexandre Richard.

In collaboration with Prof. Ely Merzbach (Bar Ilan University, Israel).

In the set-indexed framework of Ivanoff and Merzbach ( [54] ), stochastic processes can be indexed not only by $ℝ$ but by a collection $𝒜$ of subsets of a measure and metric space $(𝒯, d, m)$ , with some assumptions on $𝒜$ . In [25] , we introduce and study some assumptions on the metric indexing collection $(𝒜, d_{𝒜})$ in order to obtain a Kolmogorov criterion for continuous modifications of SI stochastic processes. Under this assumption, the collection is totally bounded and a set-indexed process with good incremental moments will have a modification whose sample paths are almost surely Hölder continuous, for the distance $d_{𝒜}$ .

Once this condition is established, we investigate the definition of Hölder coefficients for SI processes. From the real-parameter case, the most straightforward are the local (and pointwise) Hölder exponents around $U_{0} \in 𝒜$ :

{\tilde{α}}_{X} (U_{0}) = sup \{α : \underset{ρ \to 0}{lim sup} sup_{U, V \in B_{d_{𝒜}} (U_{0}, ρ)} \frac{| X_{U} - X_{V} |}{d_{𝒜} {(U, V)}^{α}} < \infty\} .

When the processes are Gaussian, a deterministic counterpart to this exponent is defined as it is in the real-parameter framework. For all $U_{0} \in 𝒜$ , we proved that almost surely, the random and the deterministic exponents are equal. Also, we proved that for the local exponents, this result holds almost surely, uniformly on $𝒜$ .

Given the particular structure of $𝒜$ , other coefficients of Hölder regularity were studied on $𝒞$ :

𝒞 = \{A ∖ ⋃_{k = 1}^{n} B_{k} : A, B_{1}, \dots, B_{n} \in 𝒜, n \in ℕ\} .

(15)

On specific subclasses $𝒞^{l}$ of $𝒞$ (satisfying $\cup_{l \geq 1} 𝒞^{l} = 𝒞$ ), the local (and pointwise) $𝒞^{l}$ -Hölder exponents are defined:

{\tilde{α}}_{X, 𝒞^{l}} (U_{0}) = sup \{α : \underset{ρ \to 0}{lim sup} sup_{\begin{matrix} U \in B_{d_{𝒜}} (U_{0}, ρ) \\ V \in ℬ^{l} (U_{0}, ρ) \end{matrix}} \frac{| Δ X_{U ∖ V} |}{d_{𝒜} {(U, V)}^{α}} < \infty\},

(16)

and this definition is proved to be independent of $l$ , leading to the definition of ${\tilde{α}}_{X, 𝒞} (U_{0})$ . It is compared to ${\tilde{α}}_{X} (U_{0})$ and related to the Hölder exponent of the process projected on flows (a flow is a continuous increasing path in $𝒜$ ). This last technique permits to show that the pointwise Hölder exponent of the SIfBm is almost surely uniformly equal to $H$ , the Hurst parameter of the SIfBm. This completes some previous results on the multiparameter fractional Brownian motion.

The last exponent which is studied is the exponent of pointwise continuity:

α_{X}^{p c} (t) = sup \{α : \underset{n \to \infty}{lim sup} \frac{| Δ X_{C_{n} (t)} |}{m {(C_{n} (t))}^{α}} < \infty\}

(17)

for all $t \in 𝒯$ , where $C_{n} (t)$ is the smaller set of $𝒞_{n}$ containing $t$ . Almost sure results are also obtained in that case. For instance, the coefficient of pointwise continuity of a SI Brownian motion equals $1 / 2 a . s .$

All these results are finally applied to the SIfBm and the SI Ornstein-Ühlenbeck process ([1] ).

Previous |

Home | Next next