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=(X t ) t𝐑 + is a stochastic process, then for all t 0 𝐑 + , a function s ' σ X,t 0 (s ' ) called the 2-microlocal frontier is defined to characterize entirely the local regularity of X at t 0 . In particular, for all s ' 𝐑 such that σ X,t 0 (s ' )0,1, it is defined as

σ X,t 0 (s ' )=supσ:lim sup ρ0 sup u,vB(t 0 ,ρ) |X u -X v | |u-v| σ ρ -s ' <.

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 = 0 t η u dW u , with ηL 2 (𝐑).

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

X t = 0 t H u dM 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 t𝐑 + , the 2-microlocal frontier of a local martingale M, with quadratic variation M, satisfies

s ' -α M,t ;σ M,t (s ' )=Σ M,t (s ' )=1 2Σ M,t 2s ' ,

where for any process X, Σ X,t denotes the pseudo 2-microlocal frontier which is characterized as following

s ' 𝐑;Σ X,t (s ' )=σ X,t (s ' )(s ' +p X,t )1,

where p X,t corresponds to

p X,t =infn 1 : X (n) (t) exists and X (n) (t) 0,

with the usual convention inf{}=+.

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 1 2 is replaced by H(t), 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 t𝐑 +

s ' -α X,t ;σ X,t (s ' )=Σ X,t (s ' )=1 2Σ 0 H u 2 dM u ,t 2s ' .

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 ωΩ and for all t𝐑 + , the 2-microlocal frontier satisfies

  1. if H t (ω)0:

    s ' 𝐑;σ X,t (s ' )=σ B,t (s ' )=1 2+s ' 1 2;
  2. if H t (ω)=0:

    s ' -α X,t ;σ X,t (s ' )=1 2+Σ H 2 ,t (2s ' ) 21 2,

    unless H is locally equal to zero at t, which induces in that case: σ X,t =+.

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 δ-dimensional Bessel processes which verify the following equation

Z t =x+2 0 t Z s dβ s +δt.