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 $s^{\text{'}}\mapsto {\sigma }_{X,t_0}\left(s^{\text{'}}\right)$ 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

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

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 $\langle M\rangle$, satisfies

where for any process $X$, ${\Sigma }_{X,t}$ denotes the pseudo 2-microlocal 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 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 $t\in {𝐑}_{+}$

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 $H_t\left(\omega \right)\ne 0$:

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

2. if $H_t\left(\omega \right)=0$:

   $\forall s^{\text{'}}\ge -{\alpha }_{X,t};{\sigma }_{X,t}\left(s^{\text{'}}\right)=\left(\frac{1}{2}+\frac{{\Sigma }_{H^2,t}\left(2s^{\text{'}}\right)}{2}\right)\wedge \frac{1}{2},$

   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