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 $U_0\in 𝒜$:

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 $𝒞$:

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

and this definition is proved to be independent of $l$, leading to the definition of ${\tilde{\alpha }}_{X,𝒞}\left(U_0\right)$. It is compared to ${\tilde{\alpha }}_X\left(U_0\right)$ 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:

for all $t\in 𝒯$, where $C_n\left(t\right)$ 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 $1/2a.s.$

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