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## Section: New Results

### Stochastic 2-microlocal analysis

Participants : Erick Herbin, Paul Balança.

Stochastic 2-microlocal analysis has been introduced in [19] to study the local regularity of stochastic processes. If $X={\left({X}_{t}\right)}_{t\in {𝐑}_{+}}$ is a stochastic process, then for all ${t}_{0}\in {𝐑}_{+}$, a function ${s}^{\text{'}}↦{\sigma }_{X,{t}_{0}}\left({s}^{\text{'}}\right)$ called the 2-microlocal frontier is defined to characterize entirely the local regularity of $X$ at ${t}_{0}$. In particular, for all ${s}^{\text{'}}\in 𝐑$ such that ${\sigma }_{X,{t}_{0}}\left({s}^{\text{'}}\right)\in \left(0,1\right)$, it is defined as

${\sigma }_{X,{t}_{0}}\left({s}^{\text{'}}\right)=sup\left\{\sigma :\underset{\rho \to 0}{lim sup}\underset{u,v\in B\left({t}_{0},\rho \right)}{sup}\frac{|{X}_{u}-{X}_{v}|}{{|u-v|}^{\sigma }{\rho }^{-{s}^{\text{'}}}}<\infty \right\}.$

The 2-microlocal 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 $\eta \in {L}^{2}\left(𝐑\right)$.

Our main goal was therefore to extend this result to any stochastic integral

${X}_{t}={\int }_{0}^{t}{H}_{u}\mathrm{d}{M}_{u},$

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 $t\in {𝐑}_{+}$, the 2-microlocal frontier of a local martingale $M$, with quadratic variation $〈M〉$, satisfies

$\forall {s}^{\text{'}}\ge -{\alpha }_{M,t};\phantom{\rule{1.em}{0ex}}{\sigma }_{M,t}\left({s}^{\text{'}}\right)={\Sigma }_{M,t}\left({s}^{\text{'}}\right)=\frac{1}{2}{\Sigma }_{〈M〉,t}\left(2{s}^{\text{'}}\right),$

where for any process $X$, ${\Sigma }_{X,t}$ denotes the pseudo 2-microlocal frontier which is characterized as following

$\forall {s}^{\text{'}}\in 𝐑;\phantom{\rule{1.em}{0ex}}{\Sigma }_{X,t}\left({s}^{\text{'}}\right)={\sigma }_{X,t}\left({s}^{\text{'}}\right)\wedge \left({s}^{\text{'}}+{p}_{X,t}\right)\wedge 1,$

where ${p}_{X,t}$ corresponds to

${p}_{X,t}=inf\left\{n\ge 1:{X}^{\left(n\right)}\left(t\right)\phantom{\rule{4.pt}{0ex}}\text{exists}\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}{X}^{\left(n\right)}\left(t\right)\ne 0\right\},$

with the usual convention $inf\left\{\varnothing \right\}=+\infty$.

As the previous result is based on Dubins-Schwarz representation theorem, it can be easily extended to characterize the regularity of time-changed 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 $t\in {𝐑}_{+}$

$\forall {s}^{\text{'}}\ge -{\alpha }_{X,t};\phantom{\rule{1.em}{0ex}}{\sigma }_{X,t}\left({s}^{\text{'}}\right)={\Sigma }_{X,t}\left({s}^{\text{'}}\right)=\frac{1}{2}{\Sigma }_{{\int }_{0}^{\begin{array}{c}•\end{array}}{H}_{u}^{2}\mathrm{d}{〈M〉}_{u},t}\left(2{s}^{\text{'}}\right).$

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 $t\in {𝐑}_{+}$, the 2-microlocal frontier satisfies

1. if ${H}_{t}\left(\omega \right)\ne 0$:

$\forall {s}^{\text{'}}\in 𝐑;\phantom{\rule{1.em}{0ex}}{\sigma }_{X,t}\left({s}^{\text{'}}\right)={\sigma }_{B,t}\left({s}^{\text{'}}\right)=\left(\frac{1}{2}+{s}^{\text{'}}\right)\wedge \frac{1}{2};$
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

${Z}_{t}=x+2{\int }_{0}^{t}\sqrt{{Z}_{s}}\mathrm{d}{\beta }_{s}+\delta t.$