Extreme-value statistics of fractional Brownian motion bridges

Fractional Brownian motion is a self-affine, non-Markovian, and translationally invariant generalization of Brownian motion, depending on the Hurst exponent $H$. Here we investigate fractional Brownian motion where both the starting and the end point are zero, commonly referred to as bridge processes. Observables are the time $t_{+}$ the process is positive, the maximum $m$ it achieves, and the time $t_{\max }$ when this maximum is taken. Using a perturbative expansion around Brownian motion ($H=\frac{1}{2}$), we give the first-order result for the probability distribution of these three variables and the joint distribution of $m$ and $t_{\max }$. Our analytical results are tested and found to be in excellent agreement, with extensive numerical simulations for both $H>\frac{1}{2}$ and $H<\frac{1}{2}$. This precision is achieved by sampling processes with a free end point and then converting each realization to a bridge process, in generalization to what is usually done for Brownian motion.

Figure 1

Examples of fBm bridges for different values of $H$, generated from the same random numbers using the Davis and Harte procedure [27]; $H=0.25$ in red (outmost curves) to $H=0.875$ in blue (innermost), with increments of $1/8$.

Figure 2

Illustration of the reflection principle: Every path emanating from 1 and attaining zero again (blue) is compensated by a reflected path emanating from $-1$ (green).

Figure 3

In red (bottom curve) a contribution to ${\tilde{W}}_1^{+}(\lambda ,s,x_1,x_2)$, where the path reaches 0 at least once (here for $x_1=0.5$ and $x_2=1$). In blue (top curve) the additional contribution to ${\tilde{W}}_2^{+}(\lambda ,s,x_1,x_2)$, where the path never reaches 0, possible when $x_1$ and $x_2$ have the same sign (here for $x_1=0.5$ and $x_2=1$).

Figure 4

Comparison of the two “arcsine laws” for a fBm bridge with Hurst exponent $H=0.66$. Dots represent the distribution extracted from numerical simulations, the plain lines represent the analytical result at order $\varepsilon$ given in Eqs. (75) and (82), and the dashed line is the scaling form (identical for both observables).

Figure 5

Left: Numerical estimation of the scaling function ${\mathcal{F}}^{\mathrm{pos}}\left(\vartheta \right)$, from top to bottom for $H=0.33$ (red dots), $H=0.4$ (orange dots), $H=0.6$ (green dots), and $H=0.66$ (blue dots), compared to the analytical result given in Eq. (76) (plane line). Right: ibid. for ${\mathcal{F}}^{\max }\left(\vartheta \right)$ for $H=0.33$ (blue dots, bottom) and $H=0.66$ (red dots, top), the analytical result (plane line) is given in Eq. (83). For both plots, and for each value of $H$, the statistics is done with $5\times {10}^6$ sampled paths, discretized with $N=2^{12}$ points.

Figure 6

Plain red line: Optimal paths for fBm conditioned to $X_0=0, X_{1/2}=1$, and $X_1=0$, for, from left to right, $H=0.1, H=0.25$, and $H=1$. The blue dashed line represents the optimal paths when neglecting the correlation between $[0,1/2]$ and $[1/2,1]$.

Figure 7

Numerical results for $P_H\left(\upsilon \right|\vartheta )$ for $H=\frac{2}{5}$ (left), $H=\frac{3}{5}$ (middle), and $H=\frac{2}{3}$ (right). The values of $\vartheta$ are chosen as $\vartheta =0, \vartheta =0.05, \vartheta =0.25$, to $\vartheta =0.5$, the maximum useful value due to the symmetry $\vartheta \to 1-\vartheta$. We used $N=2^{18}$ points and $5\times {10}^6$ samples.