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## 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 $S\left(t\right),t\in \left[0,1\right],$ is a self-stabilizing process if, for almost surely all $t\in \left[0,1\right)$, $S$ is localisable at $t$ with tangent process ${S}_{t}^{\text{'}}$ an $g\left(S\left(t\right)\right)-$stable process, with respect to the conditional probability measure ${ℙ}_{S\left(t\right)}.$ In other words,

 $\begin{array}{c}\hfill \underset{r↘0}{lim}\frac{S\left(t+ru\right)-S\left(t\right)}{{r}^{1/g\left(S\left(t\right)\right)}}={S}_{t}^{\text{'}}\left(u\right),\end{array}$ (8)

where convergence is in finite dimensional distributions with respect to ${ℙ}_{S\left(t\right)}.$ Heuristically, if ${S}_{t}^{\text{'}}\left(u\right)={L}_{g\left(S\left(t\right)\right)}\left(u\right),$ equality (8 ) implies that

$S\left(t+ru\right)-S\left(t\right)\approx {r}^{1/g\left(S\left(t\right)\right)}{L}_{g\left(S\left(t\right)\right)}\left(u\right)={\left(ru\right)}^{1/g\left(S\left(t\right)\right)}{L}_{g\left(S\left(t\right)\right)}\left(1\right),$

when $r$ is small. Thus it is natural to define $S\left(t\right)={lim}_{n\to \infty }{S}_{n}\left(\frac{⌊nt⌋}{n}\right),$ where

${S}_{n}\left(\frac{k+1}{n}\right)-{S}_{n}\left(\frac{k}{n}\right)={n}^{-1/g\left({S}_{n}\left(k/n\right)\right)}{L}_{g\left({S}_{n}\left(k/n\right)\right)}\left(1\right).$

This inspiration allows us to build Markov processes that converge to a self-stabilizing process. Note that, when $\alpha \left(x\right)\equiv 2,$ this is simply Donsker's construction. The main difficult is to prove the weak convergence of ${S}_{n}.$ To this aim, we make use of a generalization of the Arzelà-Ascoli theorem.

Definition 0.1 We call the sequence ${\left({f}_{n}\left(\theta \right)\right)}_{n\ge 1}$ is sub-equicontinuous on $I\subset {ℝ}^{d},$ if for any $\epsilon >0,$ there exist $\delta >0$ and a sequence of nonnegative numbers ${\left({\epsilon }_{n}\right)}_{n\ge 1},$ ${\epsilon }_{n}\to 0$ as $n\to \infty ,$ such that, for all functions ${f}_{n}$ in the sequence,

 $\begin{array}{c}\hfill |{f}_{n}\left({\theta }_{1}\right)-{f}_{n}\left({\theta }_{2}\right)|\phantom{\rule{4pt}{0ex}}\le \phantom{\rule{4pt}{0ex}}\epsilon +{\epsilon }_{n},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\theta }_{1},{\theta }_{2}\in I,\end{array}$ (9)

whenever $||{\theta }_{1}-{\theta }_{2}||<\delta$ (if ${\epsilon }_{n}=0$ for all $n,$ then ${\left({f}_{n}\left(\theta \right)\right)}_{n\ge 1}$ is just equicontinuous).

The slightly generalized version of the Arzelà-Ascoli theorem reads:

Lemma 0.1 Assume that ${\left({f}_{n}\right)}_{n\ge 1}$ be a sequence of real-valued continuous functions defined on a closed and bounded set ${\Pi }_{i=1}^{d}\left[{a}_{i},{b}_{i}\right]\subset {ℝ}^{d}.$ If this sequence is uniformly bounded and sub-equicontinuous, then there exists a subsequence ${\left({f}_{{n}_{k}}\right)}_{k\ge 1}$ that converges uniformly.

The following theorem states that self-stabilizing processes do exist.

Theorem 0.2 Let $g$ be a Hölder function defined on $ℝ$ and ranging in $\left[a,b\right]\subset \left(0,2\right]$. There exists a self-stabilizing process $S\left(t\right),t\in \left[0,\phantom{\rule{0.166667em}{0ex}}1\right],$ that it is tangent at all $u$ to a $g\left(S\left(u\right)\right)-$stable Lévy process under the conditional expectation with respect to $S\left(u\right)$. Moreover, the process $S\left(t\right),t\in \left[0,\phantom{\rule{0.166667em}{0ex}}1\right],$ satisfies, for all $\left({\theta }_{j},{t}_{j}\right)\in ℝ×\left[0,1\right],\phantom{\rule{0.166667em}{0ex}}j=1,2,...,d,$

 $\begin{array}{c}\hfill {𝔼}_{S\left({t}_{1}\right)}\left[exp\left\{i\sum _{j=2}^{d}{\theta }_{j}\left(S\left({t}_{j}\right)-S\left({t}_{1}\right)\right)+\int |\sum _{j=2}^{d}{\theta }_{j}{\mathbf{1}}_{\left[{t}_{1},{t}_{j}\right]}\left(z\right){|}^{g\left(S\left(z\right)\right)}dz\right\}\right]=1.\end{array}$ (10)

We are currently studying the main properties of self-stabilizing processes.