Maximum of a Fractional Brownian Motion: Analytic Results from Perturbation Theory

Fractional Brownian motion is a non-Markovian Gaussian process $X_t$, indexed by the Hurst exponent $H$. It generalizes standard Brownian motion (corresponding to $H=1/2$). We study the probability distribution of the maximum $m$ of the process and the time $t_{\max }$ at which the maximum is reached. They are encoded in a path integral, which we evaluate perturbatively around a Brownian, setting $H=1/2+ϵ$. This allows us to derive analytic results beyond the scaling exponents. Extensive numerical simulations for different values of $H$ test these analytical predictions and show excellent agreement, even for large $ϵ$.

Figure 1

Two realizations of fBM paths for different values of $H$, generated using the same random numbers for the Fourier modes in the Davis and Harte procedure [12]. The observables $m$ and $t_{\max }$ are given.

Figure 2

Graphical representation of a contribution to the path-integral $Z^{+}(m_1,t_1;x_0;m_2,t_2)$ given in Eq. (3). The red curve represents the nonlocal interaction in the action [second line of Eq. (4)] while blue lines are bare propagators. There are two other contributions when the time ordering is ${\tau }_1<{\tau }_2<t_1$ or $t_1<{\tau }_1<{\tau }_2$, already computed in Ref. [20].

Figure 3

Distribution of $t_{\max }$ for $T=1$ and $H=0.25$ (red) or $H=0.75$ (blue) given in Eq. (7) (plain lines) compared to the scaling ansatz, i.e., $\mathcal{F}=\mathrm{cst}$ (dashed lines) and numerical simulations (dots). For $H<0.5$ realizations with $t_{\max }\approx T/2$ are less probable (by about 10%) than expected from scaling. For $H>0.5$ the correction has the opposite sign.

Figure 4

Left: Numerical estimation of $\mathcal{F}$ for different values of $H$ on a discrete system of size $N=2^{12}$, using ${10}^8$ realizations. Plain curves represent the theoretical prediction (8), vertically translated for better visualization. Error bars are $2\sigma$ estimates. Note that for $H=0.6$, $H=0.66$, and $H=0.8$ the expansion parameter $ϵ$ is positive, while for $H=0.4$, $H=0.33$, and $H=0.2$ it is negative. Right: Deviation for large $|ϵ|$ between the theoretical prediction (8) and the numerical estimations (9), rescaled by $ϵ$. These curves collapse for different values of $H$, allowing for an estimate of the $O({ϵ}^2)$ correction to $P_H^T(t)$.

Figure 5

The combination (21) for $H=0.4$ (left), $H=0.6$ (middle), and $H=0.75$ (right). The plain lines are the analytical prediction $\exp (ϵ[\mathcal{G}(m/\sqrt{2})+4\ln (m)+\mathrm{cst}])$ of the maximum without its small-scale power law and large-scale Gaussian behavior. The symbols are numerical estimates for $T=1$ of the same quantity $m^{2-1/H}\exp (m^2/4)P_{\mathrm{num}}^{T=1,H}(m)$ for different sample sizes. At small scale (of order $N^{-H}$) discretization errors appear. At large scales the statistics is poor due to the Gaussian prefactor. The large scale behavior on each plot is consistent with the power law $m^{2ϵ}$.