Project-Team:REGULARITY

Inria | Raweb 2013 | Presentation of the Project-Team REGULARITY | REGULARITY Web Site


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Section: New Results

Sample path properties of multifractional Brownian motion

Participants : Paul Balança, Erick Herbin [supervision] .

In [50] , we have investigated the geometry of the sample paths of multifractional Brownian motion. Several representations of mBm exist, including the classic integral form:

\forall t \in 𝐑; X_{t} = \frac{1}{Γ (H (t) + \frac{1}{2})} \int_{𝐑} [{(t - u)}_{+}^{H (t) - 1 / 2} - {(- u)}_{+}^{H (t) - 1 / 2}] d W_{u},

where $H : 𝐑 \mapsto (0, 1)$ is a continuous function. Interestingly, we observe that geometric properties obtained in the probabilistic literature usually rely on a key assumption on the behaviour of the Hurst function:

H is a β -Hölder continuous function such that \forall t \in 𝐑, H (t) < β . (ℋ_{0})

(6)

Under the previous hypothesis, the local regularity of the mBm at $t$ corresponds to the geometry of a fractional Brownian motion of parameter $H (t)$ . Nevertheless, it has been shown in [15] that when this assumption does not hold, the sample path properties are not as simple and straightforward. More precisely, the latter has proved that the Hölder exponents satisfy at every $t \in 𝐑$ :

α_{X, t} = H (t) \land α_{H, t} and {\tilde{α}}_{X, t} = H (t) \land {\tilde{α}}_{H, t} a.s.

(7)

This result has been recently improved in [48] , observing that the pointwise exponent can even be random under some assumptions on $H$ .

Therefore, the main goal of this work was to obtain a more complete characterization of the geometry of the general mBm. We have first focused on the Hölder regularity of the sample paths, using for this purpose a deterministic representation of the fractional Brownian field:

\begin{matrix} B^{\pm} (t, H) = \frac{\pm 1}{Γ (H - \frac{1}{2})} \int_{𝐑} B_{u} [{(t - u)}_{\pm}^{H - 3 / 2} - {(- u)}_{\pm}^{H - 3 / 2}] d u, \end{matrix}

(8)

where $H \geq 1 / 2$ and $B$ is a continuous Brownian motion. Hence, observing that the mBm almost surely corresponds to the fractional integration of a Brownian motion, we have been able to use the 2-microlocal formalism and its interesting connections with fractional operators. As a consequence, we have proved that the pointwise exponent of the mBm almost surely satisfies:

\forall t \in 𝐑; α_{X, t} = H (t) \land m_{t, H (t)} α_{H, t},

(9)

where $m_{t, H (t)}$ is defined as the multiplicity of the fractional Brownian field at $(t, H)$ , i.e.

m_{t, H} = inf \{k \in 𝐍 ∖ {0} : \partial_{H}^{k} B (t, H) \neq 0\} .

We have also been able to obtain some uniform lower bounds on the 2-microlocal frontier, which are optimal under some mild assumptions on the Hurst function.

The second direction of our study has concerned the fractal dimension of the graph of the mBm. Interestingly, and on the contrary to fBm, we have to distinguish the Box and Hausdorff dimensions in our result. The first happens to be the easiest one to study and is closely related to the geometry of $H$ itself. Therefore, with probability one,

\forall t \in 𝐑 ∖ {0}; {dim}_{B, t} Gr (X) = (2 - H (t)) \lor {dim}_{B, t} Gr (H),

(10)

where ${dim}_{B, t}$ denotes the localized Box dimension at $t$ .

To study the Hausdorff dimension the graph, we need a slightly different approach which makes use of parabolic Hausdorff dimension. We first define for all $t \in 𝐑$ a parabolic metric $ϱ_{H}$ on $𝐑^{2}$ , with $H > 0$ : $ϱ_{H} ((u, x); (v, y)) : = max ({| u - v |}^{H}, | x - y |)$ . For any set $A \subset 𝐑^{2}$ , we denote by ${dim}_{ℋ} (A; ϱ_{H})$ the parabolic Hausdorff dimension of $A$ . It is defined similarly to the classic Hausdorff dimension using covering balls relatively to the metric $ϱ_{H}$ , i.e. it corresponds to the infimum of $s \geq 0$ for which

lim_{δ \to 0} inf \{\sum_{i = 0}^{\infty} diam {(O_{i}; ϱ_{H})}^{s} : {(O_{i})}_{i \in N} is a δ -cover of A\} < \infty

Studying the local Hausdorff dimension of the graph of the mBm, we have proved that with probability one

\forall t \in 𝐑 ∖ {0}; {dim}_{H, t} Gr (X) = 1 + H (t) ({dim}_{H, t} (Gr (H); ϱ_{H (t)}) - 1) .

(11)

Even though this result might seem counter-intuitive, it can be checked that it induced the classic equality ${dim}_{H, t} Gr (X) = 2 - H (t)$ when the mBm satisfies the assumption $ℋ_{0}$ . Interestingly, we observe that a similar expression has also emerged recently in the study [70] of the Hausdorff dimension of a fractional Brownian motion with variable drift. Finally, we also note this result can also been extended to images of fractal sets by the multifractional Brownian motion.

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