Section: New Results

Self-regulated processes

Participants : Jacques Lévy Véhel, Anne Philippe, Caroline Robet

We wish to construct various instances of processes Z such that, at each point t, almost surely, the pointwise Hölder exponent of Z at t, denoted ${\alpha }_Z\left(t\right)$, verifies

where $g\in {𝒞}^1(ℝ,[a,b])$ is a deterministic function. Then, we would estimate the function g which control the regularity.

The pointwise Hölder exponent at t of a function or a process $f:ℝ\to ℝ$, which is ${𝒞}^1$ nowhere, is the real ${\alpha }_f\left(t\right)$ such that :

We worked first on pathwise integrals :

Theorem 1 Let $g\in {𝒞}^1(ℝ,[a,b])$, $0<a<b<1$. Provided $\parallel g^{\text{'}}{\parallel }_{\infty }$ is small enough, there exists a unique continuous process Z verifying almost surely on $[0,T]$

where W is an almost surely continuous process.

A random condition ($\parallel g^{\text{'}}{\parallel }_{\infty }{\parallel W\left(\omega \right)\parallel }_{\infty }C(a,T)<1$) appears in the application of Banach fixed point theorem (in $({𝒞}^0([0,T];ℝ),\parallel .{\parallel }_{\infty })$). It implies that it is possible to have existence et uniqueness only on $[0,t^{\text{'}}]$, $t^{\text{'}}<T$. We simulated pathwise integrals and showed some cases without uniqueness. We studied some easier processes in order to find the regularity of Z.

Theorem 2 Let $h\in ]0,1[$ and U defined on $[0,T]$ by

Then $\forall t\in [0,T]$, ${\alpha }_U\left(t\right)\ge h$.

Theorem 3 Let $g\in {𝒞}^1(ℝ,[a,b])$, $0<a<b<1$. Provided $\parallel g^{\text{'}}{\parallel }_{\infty }$ is small enough, there exists a unique continuous process Y verifying almost surely on $[0,T]$

where W is an almost surely continuous process. Furthermore, $\forall t\in [0,T]$, ${\alpha }_Y\left(t\right)\ge g\left(Y_t\right)$

Then, we adapted the multifractional Brownian Motion [50], [31] (which a representation is $B_t={\int }_0^t{K}_{H\left(t\right)}(t,u)W\left(du\right)$, W Brownian Motion et H$\in {𝒞}^1$) to construct the modified multifractional Brownian Motion : $Z_t={\int }_0^t{K}_{H\left(u\right)}(t,u)W\left(du\right)$. We expect obtain a self-regulated process $Y_t={\int }_0^t{K}_{g\left(Y_u\right)}(t,u)dW\left(u\right)$.

Theorem 4 Let $g\in {𝒞}^1(ℝ,[a,b])$, $0<a<b<1$. Provided $\parallel g^{\text{'}}{\parallel }_{\infty }$ is small enough, there exists a unique continuous adapted process Y include in ${𝒞}^0([0,T];L^2\left(\Omega \right))$ verifying almost surely on $[0,T]$

where W is the Brownian motion.