[Paper Review] Dimension of Fractional Brownian motion with variable drift

Yuval Peres, Perla Sousi|arXiv (Cornell University)|Oct 25, 2013

Stochastic processes and financial applications5 references4 citations

TL;DR

This paper establishes explicit formulas for the Hausdorff dimension of the graph and image of fractional Brownian motion perturbed by a Borel measurable function $ f $, using a novel adaptation of parabolic Hausdorff dimension. The key result shows that the dimension of $ X + f $ can strictly exceed both the dimension of $ X $ and $ f $, even for Hölder-continuous $ f $, and that Minkowski and Hausdorff dimensions may differ despite random perturbation.

ABSTRACT

Let $X$ be a fractional Brownian motion in $\mathbb{R}^d$. For any Borel function $f:[0,1] o \mathbb{R}^d$, we express the Hausdorff dimension of the image and the graph of $X+f$ in terms of $f$. This is new even for the case of Brownian motion and continuous $f$, where it was known that this dimension is almost surely constant. The expression involves an adaptation of the parabolic dimension previously used by Taylor and Watson to characterize polarity for the heat equation. In the case when the graph of $f$ is a self-affine McMullen-Bedford carpet, we obtain an explicit formula for the dimension of the graph of $X+f$ in terms of the generating pattern. In particular, we show that it can be strictly bigger than the maximum of the Hausdorff dimension of the graph of $f$ and that of $X$. Despite the random perturbation, the Minkowski and Hausdorff dimension of the graph of $X+f$ can disagree.

Motivation & Objective

• To determine the almost sure Hausdorff dimension of the graph and image of $ X + f $, where $ X $ is fractional Brownian motion and $ f $ is a Borel function.
• To resolve whether the Hausdorff dimension of $ B + f $ can strictly exceed $ \max\{\dim(\mathrm{Gr}(B)), \dim(\mathrm{Gr}(f))\} $ for standard Brownian motion $ B $.
• To investigate whether strict inequality between Minkowski and Hausdorff dimensions in self-affine graphs persists under random perturbations by Brownian motion.
• To provide explicit dimension formulas for graphs of self-affine functions constructed via McMullen-Bedford carpets under fractional Brownian motion perturbation.

Proposed method

• Introduces an adaptation of parabolic Hausdorff dimension, defined via $ H $-parabolic $ \beta $-dimensional content over rectangles with time and space scales $ \delta $ and $ \delta^H $.
• Uses the $ H $-parabolic Hausdorff dimension of $ \mathrm{Gr}_A(f) $, denoted $ \dim_{\Psi,H}(\mathrm{Gr}_A(f)) $, as the key parameter to express the dimension of $ \mathrm{Gr}_A(X+f) $ and $ \mathcal{R}_A(X+f) $.
• Applies the formula $ \dim(\mathrm{Gr}_A(X+f)) = \min\{\alpha/H, \alpha + d(1-H)\} $ and $ \dim(\mathcal{R}_A(X+f)) = \min\{\alpha/H, d\} $, where $ \alpha = \dim_{\Psi,H}(\mathrm{Gr}_A(f)) $.
• Analyzes self-affine McMullen-Bedford carpets by computing the parabolic dimension of their graphs using recursive constructions and ergodic averages of measures on dyadic rectangles.
• Applies the strong law of large numbers to compute the parabolic dimension of the graph of a Hölder-continuous function $ f $ with exponent $ \theta = \log 2 / \log 6 $.
• Uses Cauchy-Schwarz and Jensen’s inequality to compare the Hausdorff and Minkowski dimensions of $ \mathrm{Gr}(B+f) $, showing strict inequality when row counts $ r_j $ are unequal.

Experimental results

Research questions

• RQ1 Can the Hausdorff dimension of the graph of $ B + f $ strictly exceed the maximum of the dimensions of $ B $ and $ f $, where $ B $ is standard Brownian motion?
• RQ2 Does the Minkowski dimension of $ \mathrm{Gr}(B+f) $ remain equal to that of $ \mathrm{Gr}(f) $ when $ f $ is a self-affine function with unequal row counts?
• RQ3 Can the parabolic Hausdorff dimension of $ f $ be computed explicitly for self-affine graphs, and does it yield a strictly larger dimension for $ \mathrm{Gr}(X+f) $ than either $ X $ or $ f $ alone?
• RQ4 Is the strict inequality between Minkowski and Hausdorff dimensions of $ \mathrm{Gr}(f) $ preserved under Brownian perturbation?

Key findings

• For a Hölder-continuous function $ f $ with exponent $ \theta = \log 2 / \log 6 $, the parabolic Hausdorff dimension of $ \mathrm{Gr}(f) $ is $ \frac{1}{2} \log_2(5^{2\theta} + 1) $.
• The Hausdorff dimension of $ \mathrm{Gr}(B + f) $ is $ \frac{\log_2(5^{2\theta} + 1) + 1}{2} $, which strictly exceeds both $ \dim(\mathrm{Gr}(f)) = \log_2(5^\theta + 1) $ and $ \dim(\mathrm{Gr}(B)) = 3/2 $.
• For a self-affine graph $ \mathrm{Gr}(f) $ with pattern $ D $, the dimension of $ \mathrm{Gr}(X + f) $ is $ 1 - H + H \log_m \left( \sum_{j=0}^{m-1} r(j)^{\log_n m / H} \right) $, with $ H < \log_n m $.
• When the row counts $ r_j $ are not all equal, the Minkowski dimension of $ \mathrm{Gr}(B+f) $ equals that of $ \mathrm{Gr}(f) $, but the Hausdorff dimension is strictly larger than both $ \dim(\mathrm{Gr}(B)) $ and $ \dim(\mathrm{Gr}(f)) $.
• Despite the random perturbation, the Minkowski and Hausdorff dimensions of $ \mathrm{Gr}(B+f) $ can disagree, with the Minkowski dimension being strictly larger.
• The dimension of the image $ \mathcal{R}_A(X+f) $ is $ \min\{\alpha/H, d\} $, where $ \alpha $ is the parabolic dimension of $ \mathrm{Gr}_A(f) $.