Perturbative expansion for the maximum of fractional Brownian motion

Brownian motion is the only random process which is Gaussian, scale invariant, and Markovian. Dropping the Markovian property, i.e., allowing for memory, one obtains a class of processes called fractional Brownian motion, indexed by the Hurst exponent $H$. For $H=1/2$, Brownian motion is recovered. We develop a perturbative approach to treat the nonlocality in time in an expansion in $\varepsilon =H-1/2$. This allows us to derive analytic results beyond scaling exponents for various observables related to extreme value statistics: the maximum $m$ of the process and the time $t_{\max }$ at which this maximum is reached, as well as their joint distribution. We test our analytical predictions with extensive numerical simulations for different values of $H$. They show excellent agreement, even for $H$ far from $1/2$.

Figure 1

The three arcsine laws discussed in the text. $t_{\max }$ (vertical green line) is the time at which the process achieves its maximum. $t_{\mathrm{last}}$ (vertical blue line) is the last time at which the process is at its starting value $X_0=0$. Finally, $t_{+}$(horizontal red lines) is the time spent in the positive half-space, which is the sum of the red intervals.

Figure 2

Two realizations of fBm paths for different values of $H$, generated using the same random numbers for Fourier modes in the Davis and Harte procedure [23]. The observables $m$ and $t_{\max }$ are shown.

Figure 3

(a) Graphical representation of the contribution $Z_{\gamma }^{+}$ to the path integral $Z^{+}(m_1,t_1;x_0;m_2,t_2)$ given in Eq. (12). The red curve represents the nonlocal interaction in the action [second line of Eq. (14)], while blue lines are bare propagators. We also indicate the Laplace variable which appears in each time slice in Eq. (32). (b) Graphical representation of $Z_{\alpha }^{+}$.

Figure 4

Distribution of $t_{\max }$ for $T=1$ and $H=0.25$ (red curves) or $H=0.75$ (blue curves) given in Eq. (51) (solid lines) compared to the scaling ansatz, i.e., $\mathcal{F}=\text{cst.}$ (dashed lines) and numerical simulations (dotted lines). 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 5

Scaling function $f_H\left(y\right)$ for the distribution of the maximum, as defined in Eq. (35), for different values of $H$: $H=0.25$ (red curve), $H=0.4$ (yellow curve), $H=0.6$ (green curve), and $H=0.75$ (blue curve). Solid lines represent the analytic prediction from our perturbative theory (at first order in $\varepsilon$) given in Eq. (55); symbols are results from numerical simulations (cf. Sec. 5).

Figure 6

Survival probability $S\left(y\right)$ for $H=1/2$ (solid blue line), $H=0.75$ (dashed red line), $H=0.25$ (dot-dashed green line), and asymptotics $S\left(y\right)=1$ (dotted black line), in a log-log plot.

Figure 7

(a) Conditional probability $P_H\left(y\right|\vartheta )$ for $H=\frac{2}{3}$ and various values of $\vartheta$. (b), (c) Same as (a), for $H=\frac{3}{5}$ and $H=\frac{1}{3}$. Solid curves are the analytical prediction, (70), where the scaling functions are given analytically for the two extremal cases, $\vartheta =0$ and $\vartheta =1$ [Eqs. (72) and (73)]; for $0<\vartheta <1$ the curves are obtained via numerical integration. The predicted spread of the curves [which collapse for $H=\frac{1}{2}$ to Eq. (66), plotted with black circles) is well reproduced in the numerics, for both $\varepsilon >0$ and $\varepsilon <0$. For $\vartheta \to 1$ the agreement with numerics is remarkable, while for $\vartheta$ close to 0, we see significant deviations. These deviations may be due to both discretization effects and ${\varepsilon }^2$ corrections (they have the same sign for both $\varepsilon >0$ and $\varepsilon <0$).

Figure 8

(a) Numerical estimation of $\mathcal{F}$ for different values of $H$ in a discrete system of size $N=2^{12}$, using ${10}^8$ realizations. Solid curves represent the theoretical prediction, (52), 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 $\varepsilon$ is positive, while for $H=0.4, H=0.33$, and $H=0.2$ it is negative. (b) Deviation for large $\left|\varepsilon \right|$ between the theoretical prediction, (52), and the numerical estimations, (77), rescaled by $\varepsilon$ [cf. Eq. (78)]. These curves collapse for different values of $H$, allowing for an estimate of the $O\left({\varepsilon }^2\right)$ correction to $P_H^T\left(t\right)$, as written in Eq. (79).

Figure 9

(b) The combination, (81), for $H=0.6$. The solid line is the analytical prediction $exp(\varepsilon [\mathcal{G}(m/\sqrt{2})+4lnm]+\text{cst})$ of the distribution of the maximum without its small-scale power law and large-scale Gaussian behavior. Symbols are numerical estimations for $T=1$ of the same quantity, $m^{2-1/H}exp\left(m^2/4\right)P_{\mathrm{num}}^{T=1,H}\left(m\right)$, for different sample sizes. At small scales discretization errors appear. At large scales the statistics is poor due to the Gaussian prefactor. For the four decades in between, theory and numerics are in very good agreement. (a) Same as (b), for $H=0.4$; (c) same as (b), for $H=0.75$. In all cases, the large scale-behavior in both plots is consistent with $m^{2\varepsilon }$.