Section: New Results
Stochastic 2microlocal analysis
Participants : Erick Herbin, Paul Balança.
Stochastic 2microlocal analysis has been introduced in [19] to study the local regularity of stochastic processes. If $X={\left({X}_{t}\right)}_{t\in {\mathbf{R}}_{+}}$ is a stochastic process, then for all ${t}_{0}\in {\mathbf{R}}_{+}$, a function ${s}^{\text{'}}\mapsto {\sigma}_{X,{t}_{0}}\left({s}^{\text{'}}\right)$ called the 2microlocal frontier is defined to characterize entirely the local regularity of $X$ at ${t}_{0}$. In particular, for all ${s}^{\text{'}}\in \mathbf{R}$ such that ${\sigma}_{X,{t}_{0}}\left({s}^{\text{'}}\right)\in \left(0,1\right)$, it is defined as
The 2microlocal frontier gives a more complete picture of the regularity than classical pointwise and local Hölder exponents, which are widely used in the literature. Furthermore, it is stable under the action of (pseudo)differential operators.
[19] mainly focused on Gaussian processes, and in particular obtained a characterization of the regularity for Wiener integrals ${X}_{t}={\int}_{0}^{t}{\eta}_{u}\mathrm{d}{W}_{u}$, with $\eta \in {L}^{2}\left(\mathbf{R}\right)$.
Our main goal was therefore to extend this result to any stochastic integral
where $M$ is a local martingale and $H$ an adapted continuous process.
In fact, in [15] , we first reduced this problem to the study of local martingales, and we have shown that almost surely for all $t\in {\mathbf{R}}_{+}$, the 2microlocal frontier of a local martingale $M$, with quadratic variation $\langle M\rangle $, satisfies
where for any process $X$, ${\Sigma}_{X,t}$ denotes the pseudo 2microlocal frontier which is characterized as following
where ${p}_{X,t}$ corresponds to
with the usual convention $inf\left\{\varnothing \right\}=+\infty $.
As the previous result is based on DubinsSchwarz representation theorem, it can be easily extended to characterize the regularity of timechanged multifractional Brownian motions. In this case, we obtain a similar equation where $\frac{1}{2}$ is replaced by $H\left(t\right)$, the value of the Hurst function at $t$.
Using this last equality, we can obtain the regularity of the stochastic integral $X$ previously defined: almost surely for all $t\in {\mathbf{R}}_{+}$
In the particular case of an integration with respect to a Brownian motion $B$, the result can be simplified using the stability under differential operators: for almost all $\omega \in \Omega $ and for all $t\in {\mathbf{R}}_{+}$, the 2microlocal frontier satisfies

if ${H}_{t}\left(\omega \right)\ne 0$:
$\forall {s}^{\text{'}}\in \mathbf{R};\phantom{\rule{1.em}{0ex}}{\sigma}_{X,t}\left({s}^{\text{'}}\right)={\sigma}_{B,t}\left({s}^{\text{'}}\right)=\left(\frac{1}{2}+{s}^{\text{'}}\right)\wedge \frac{1}{2};$ 
if ${H}_{t}\left(\omega \right)=0$:
$\forall {s}^{\text{'}}\ge {\alpha}_{X,t};\phantom{\rule{1.em}{0ex}}{\sigma}_{X,t}\left({s}^{\text{'}}\right)=\left(\frac{1}{2}+\frac{{\Sigma}_{{H}^{2},t}\left(2{s}^{\text{'}}\right)}{2}\right)\wedge \frac{1}{2},$unless $H$ is locally equal to zero at $t$, which induces in that case: ${\sigma}_{X,t}=+\infty $.
Based on this last characterization, we were able to study the regularity of stochastic diffusions. In particular, we illustrated our purpose with the square of $\delta $dimensional Bessel processes which verify the following equation