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 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 𝒜:

α ˜ X (U 0 )=supα:lim sup ρ0 sup U,VB d 𝒜 (U 0 ,ρ) |X U -X V | d 𝒜 (U,V) α <.

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 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 ,,B n 𝒜,n.(15)

On specific subclasses 𝒞 l of 𝒞 (satisfying l1 𝒞 l =𝒞), the local (and pointwise) 𝒞 l -Hölder exponents are defined:

α ˜ X,𝒞 l (U 0 )=supα:lim sup ρ0 sup UB d 𝒜 (U 0 ,ρ)V l (U 0 ,ρ) |ΔX UV | d 𝒜 (U,V) α <,(16)

and this definition is proved to be independent of l, leading to the definition of α ˜ X,𝒞 (U 0 ). It is compared to 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 pc (t)=supα:lim sup n |ΔX C n (t) | m(C n (t)) α <(17)

for all t𝒯, 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 1/2a.s.

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