First passage in an interval for fractional Brownian motion

Let $X_t$ be a random process starting at $x\in [0,1]$ with absorbing boundary conditions at both ends of the interval. Denote by $P_1\left(x\right)$ the probability to first exit at the upper boundary. For Brownian motion, $P_1\left(x\right)=x$, which is equivalent to $P_1^{\prime }\left(x\right)=1$. For fractional Brownian motion with Hurst exponent $H$, we establish that $P_1^{\prime }\left(x\right)=\mathcal{N}{\left[x(1-x)\right]}^{1/H-2}{e}^{\epsilon \mathcal{F}\left(x\right)+O\left({\epsilon }^2\right)}$, where $\epsilon =H-\frac{1}{2}$. The function $\mathcal{F}\left(x\right)$ is analytic and well approximated by its Taylor expansion $\mathcal{F}\left(x\right)\simeq 16(C-1){(x-\frac{1}{2})}^2+O{(x-\frac{1}{2})}^4$, where $C=0.915...$ is the Catalan constant. A similar result holds for moments of the exit time starting at $x$. We then consider the span of $X_t$, i.e., the size of the (compact) domain visited up to time $t$. For Brownian motion, we derive an analytic expression for the probability that the span reaches 1 for the first time and then generalize it to fractional Brownian motion. Using large-scale numerical simulations with system sizes up to $N=2^{24}$ and a broad range of $H$, we confirm our analytic results. There are important finite-discretization corrections which we quantify. They are most severe for small $H$, necessitating going to the large systems mentioned above.

Figure 1

Realizations of a FBM for $H=0.25$ (red, roughest curve); $H=0.375$ (orange); $H=1/2$, Brownian (black); $H=0.625$ (green); $H=0.75$ (cyan); $H=0.875$ (bright blue); and $H=1$ (dark blue, straight line), using the algorithm of Davies and Harte [10, 11, 12].

Figure 2

Scaling function $\mathcal{F}\left(x\right)$ defined in Eq. (72) (solid red curve), normalized such that ${\int }_0^1\mathcal{F}\left(x\right)dx=0$. The dashed line is the quadratic term given in Eq. (8).

Figure 3

Effective measured curvature $\gamma$, as a function of $H$ and system size. The gray dashed line is $\gamma$ as given by Eq. (118); the black solid line at $-1.39$ is a fit to the curvature of $\mathcal{F}\left(x\right)$ in the range 0.15–0.85. The color code is [from bottom to top in (a)] $N=2^{13}$ (dark green), $2^{14}$ (green), $2^{16}$ (olive), $2^{18}$ (orange), $2^{20}$ (red), $2^{22}$ (dark magenta), and $2^{24}$ (blue). The 1-$\sigma$ errors are indicated with bars in (a) and are explicitly shown in (b). There the solid lines are the measured errors obtained as follows: $\gamma$ is estimated by fitting a parabola to each of the data sets presented in Fig. 12 and measuring the variance of the data minus the fit, which allows one to estimate the error of the fit, after calibration on white noise. The dashed lines are proportional to the square root of the number of samples, divided by $\left|\epsilon \right|$, and calibrated against the estimated data to have a second independent estimate. The gray lines are a guide to the eye, at the location of the $H$ values considered, $H=0.33$, 0.4, 0.45, 0.475, 0.525, 0.55, 0.6, 0.67, and 0.75.

Figure 4

(a) Probability that the span reaches 1 for the first time. Gray shading indicates the random-walk (RW) simulation with $\delta t={10}^{-5}$ and ${10}^6$ samples, the green curve the analytic result (39), the orange dashed curve the small-time asymptote (41), and the blue dashed curve the large-time asymptote (42). Note a small systematic deviation due to the relatively large time step $\delta t={10}^{-5}$. (b) Same as (a) on a logarithmic scale.

Figure 5

(a) Density of the span at time $t=1$. Gray shading indicates the RW simulation with $\delta t={10}^{-4}$ and ${10}^6$ samples, the green curve the analytic result (47), the orange dashed curve the small-$s$ asymptotics (48), and the blue dashed curve the large-$s$ asymptotics (49). (b) Same as (a) but on a logarithmic scale.

Figure 6

Graphic representation of the path integral for the order-$\epsilon$ contribution $\tilde{\mathcal{A}}(x,s)$ given in Eq. (60).

Figure 7

(a) Function ${\mathcal{F}}_T\left(x\right)$, defined in Eq. (85). (b) Function ${\mathcal{F}}_{T^2}\left(x\right)$, defined in Eq. (89). The dots are the numerically obtained points and the line is the fit of Eq. (90).

Figure 8

Ratio $R\left(x\right)$ defined in Eq. (96). Shown in gray are the analytical (parameter-free) predictions and in red or blue are the data points of numerical simulations, presented in Sec. 5. The red data points have been used to extract via a polynomial fit of order 40 (dashed green line) $R\left(0\right)$, with the result $R_{H=0.45}^{\mathrm{num}}\left(0\right)=0.478$ and $R_{H=0.55}^{\mathrm{num}}\left(0\right)=0.522$. Analytically, Eq. (97) yields $R_{H=0.45}^{\mathrm{ana}}\left(0\right)=0.479$ and $R_{H=0.55}^{\mathrm{ana}}\left(0\right)=0.521$. The deviation in the middle of the domain is of order $7\times {10}^{-3}$, consistent with an $O\left({\epsilon }^2\right)$ correction to $R\left(x\right)$ of amplitude 3. More robust tests of our formulas, focusing on the shape of $R\left(x\right)$ and the spatially averaged exit times, are presented in Sec. 5c.

Figure 9

(a) Random process $X_t$, with its running maximum (in red) and minimum (in blue) (see the text). (b) Span, i.e., running maximum minus running minimum.

Figure 10

Measured probability $P_1^{\prime }\left(x\right)$ for $H=0.33$. The system sizes are (from bottom to top at $x=0.5$) $N=2^{13}$ (dark green), $2^{14}$ (green), $2^{16}$ (olive), $2^{18}$ (orange), $2^{20}$ (red), $2^{22}$ (dark magenta), and $2^{24}$ (blue). The dashed line is the scaling ansatz (70) (i.e., almost a parabola). Note the slow convergence for $x\to 0$ and $x\to 1$. Also note that the measured result for the largest system size at $x=1/2$ is larger than the scaling ansatz (70). This is equivalent to a positive curvature of the function $\mathcal{F}\left(x\right)$ defined in Eqs. (71) and (72) and given in Fig. 12.

Figure 11

Number of samples for each system size and value of $H$. The color code is as in Fig. 10. Since the measured signal is proportional to $\epsilon$, the error scales like the square root of the number of samples divided by $\left|\epsilon \right|$. Thus more samples are needed for $H$ close to $1/2$, and only small systems can be simulated for $H=0.475$ and $H=0.525$.

Figure 12

(a)-(h) Numerically estimated scaling function $\mathcal{F}\left(x\right)$ for (from top to bottom) $N=2^{13}$ (dark green), $2^{14}$ (green), $2^{16}$ (olive), $2^{18}$ (orange), $2^{20}$ (red), $2^{22}$ (dark magenta), and $2^{24}$ (blue). The black dashed line is the result of Eq. (72). For $\epsilon =\pm 0.025$, i.e., $H=0.475$ and $H=0.525$, due to the large statistics needed, we only simulated systems up to size $N=2^{20}$. For $H\ge 0.6$, convergence in system size is good, and we skipped the largest system $N=2^{24}$. Note that the scaling ansatz of (70), i.e., ${\mathcal{P}}_{1,\mathrm{scaling}}^{\prime }\left(x\right)\sim {\left[x(1-x)\right]}^{1/H-2}$ is equivalent to $\mathcal{F}\left(x\right)\equiv 0$.

Figure 13

(a)-(f) Dependence of the scaling function $\mathcal{F}\left(x\right)$ on $N$, by taking the mean between $H=1/2\pm \epsilon$, for $H=0.33/0.67$ (green), $H=0.4/0.6$ (red), $H=0.45/0.55$ (blue), and $H=0.475/0.525$ (olive and yellow). For $N=2^{24}$, we have replaced the estimate for $H\ge 0.6$ by those of $N=2^{22}$, justified by the much better convergence in system size for these values of $H$ (see Fig. 12). Shown in (g) and (h) is the extracted curvature $\gamma$ with system sizes increasing from bottom to top: $N=2^{13}$ (green), $N=2^{14}$ (bright green), $N=2^{16}$ (olive and yellow), $N=2^{18}$ (orange), $N=2^{20}$ (red), $N=2^{22}$ (violet), and $N=2^{24}$ (blue, single dot with big error bars). For the error estimate see Fig. 3. Note that the scaling ansatz of (70), i.e., ${\mathcal{P}}_{1,\mathrm{scaling}}^{\prime }\left(x\right)\sim {\left[x(1-x)\right]}^{1/H-2}$, is equivalent to $\mathcal{F}\left(x\right)\equiv 0$.

Figure 14

(a) Mean (red) between the measured functions ${\mathcal{F}}_T\left(x\right)$ for $H=0.45$ (dark cyan, bottom line) and $H=0.55$ (dark green, top line). (b) Same as (a) but for ${\mathcal{F}}_{T^2}\left(x\right)$. The system size used is $N=2^{24}$ and the total number of samples is $1.64\times {10}^6$ for $H=0.45$ and $2.19\times {10}^6$ for $H=0.55$.

Figure 15

(a) Mean, i.e., spatially averaged absorption time $\langle T_{\mathrm{abs}}\rangle ={\int }_0^{1}dx\langle T_{\mathrm{abs}}\left(x\right)\rangle$. (b) Same as (a) but for the second moment. The black dotted line is the direct expansion in $\epsilon$, which underestimates the true result. The dashed line is the exponentiation of this correction; it overestimates the result. The fit to a quadratic polynomial is given in Eqs. (121) and (122).

Figure 16

(a) Expectation $\langle T_1\rangle$. The data points with interpolation follow the color code of Fig. 17. The black dotted line from Eq. (108) is the direct expansion in $\epsilon$, which underestimates the numerical result, while its exponentiation given by the dashed line overestimates it. (b) Same as (a) but for $\langle T_1^2\rangle$, with the analytical result given in Eq. (109).

Figure 17

Probability distribution of the numerically estimated time $T_1$ when the width of the process reaches 1, i.e., the process is absorbed irrespective of its starting position, for $N=2^{13}$ (dark green), $2^{14}$ (green), $2^{16}$ (olive), $2^{18}$ (orange), $2^{20}$ (red), $2^{22}$ (dark magenta), and $2^{24}$ (blue). The characteristic time depends quite strongly on $H$. The result for $H=1/2$ is between the distributions for $H=0.475$ and $H=0.525$. The black dotted line is the analytic result for $H=1/2$, given in Eqs. (39, 40), rescaled so that the first moment $\langle T_1\rangle$ is correctly reproduced. Small systematic deviations are visible for small and large values of $H$, especially $H=0.33$ and $H=0.75$.

Figure 18

Same as Fig. 17 but on a logarithmic scale. The little bump at large $T$ corresponds to all realizations which did not exit up to that time, i.e., it is the integrated tail.

Figure 19

The top seven curves are the logarithm of ${\tilde{P}}^{\prime }\left(\alpha \right)$ [Laplace transform of $P^{\prime }\left(x\right)$, as plotted in Fig. 10] for system sizes $N=2^{13}$ (top, green) to $N=2^{24}$ (blue). The lower solid curves are a scaling collapse of all seven curves, using Eqs. (132) and (133). The blue dashed line is the Laplace transform of the scaling ansatz (70), plotted in Fig. 10.