Local Hölder regularity for set-indexed processes

Abstract

International audienceIn this paper, we study the Hölder regularity of set-indexed stochastic processes defined in the framework of Ivanoff-Merzbach. The first key result is a Hölder-continuity Theorem derived from the approximation of the indexing collection by a nested sequence of finite subcollections. Hölder-continuity based on the increment definition for set-indexed processes is also considered. Then, the localization of these properties leads to various definitions of Hölder exponents. Moreover, a pointwise continuity exponent is defined in relation with the weak continuity property for set-indexed processes which only considers single point jumps. In the case of Gaussian processes, almost sure values are proved for the Hölder exponents. As an application, the local regularity of the set-indexed fractional Brownian motion and the Ornstein-Uhlenbeck process are proved to be constant, with probability one

A fractional Brownian field indexed byL2and a varying Hurst parameter

“…It is possible to establish precise Hölder regularity coefficients. For easier comparison with prior works on mpfBm, we will not use the distance d m but a variant defined as: [19] states that this distance is equivalent to the Euclidean distance when m is the Lebesgue measure and the set of indexing points stays within a compact away from 0.…”

“…The first step is to evaluate α B h (t 0 ) and α B h (t 0 ). A result of [19] then states that a Gaussian process X , indexed by a collection of sets satisfying certain technical assumptions has the following property:…”

“…As discussed in the aforementioned paper, the technical assumptions are satisfied by the class of rectangles. A result of that sort actually originated in [17], but we use the one in [19] to introduce the extended results of the following section on set-indexed processes.…”

“…Proposition 5.5. Let be an indexing collection satisfying Assumption 1 of [19]. Let B h be a regular SImBm on .…”

“…We take a closer look at the Hölder regularity of the fBf in the fifth section, when the L 2 indexing collection is restricted to the indicator functions of the rectangles of R d (multiparameter processes) or to some indexing collection (in the sense of [21]). This restriction permits to use local Hölder regularity exponents, in the flavour of what was done in [19]. When a regular path h : L 2 → (0, 1/2] is specified, this defines a multifractional Brownian field as B h f = B h( f ), f , whose Hölder regularity at each point is proved to equal h( f ) almost surely.…”