Section: New Results
Self-stabilizing Lévy motions
Participants : Xiequan Fan, Jacques Lévy Véhel.
Self-stabilizing processes have the property that the “local intensities of jumps” varies with amplitude. They are good models for, e.g., financial and temperature records.
The main aim of our work is to establish the existence of such processes and to give a simple construction. Formally, one says that a stochastic process is a self-stabilizing process if, for almost surely all , is localisable at with tangent process an stable process, with respect to the conditional probability measure In other words,
where convergence is in finite dimensional distributions with respect to Heuristically, if equality (8 ) implies that
when is small. Thus it is natural to define where
This inspiration allows us to build Markov processes that converge to a self-stabilizing process. Note that, when this is simply Donsker's construction. The main difficult is to prove the weak convergence of To this aim, we make use of a generalization of the Arzelà-Ascoli theorem.
Definition 0.1 We call the sequence is sub-equicontinuous on if for any there exist and a sequence of nonnegative numbers as such that, for all functions in the sequence,
whenever (if for all then is just equicontinuous).
The slightly generalized version of the Arzelà-Ascoli theorem reads:
Lemma 0.1 Assume that be a sequence of real-valued continuous functions defined on a closed and bounded set If this sequence is uniformly bounded and sub-equicontinuous, then there exists a subsequence that converges uniformly.
The following theorem states that self-stabilizing processes do exist.
Theorem 0.2 Let be a Hölder function defined on and ranging in . There exists a self-stabilizing process that it is tangent at all to a stable Lévy process under the conditional expectation with respect to . Moreover, the process satisfies, for all
We are currently studying the main properties of self-stabilizing processes.