Sampling first-passage times of fractional Brownian motion using adaptive bisections

Abstract

We present an algorithm to efficiently sample first-passage times for fractional Brownian motion. To increase the resolution, an initial coarse lattice is successively refined close to the target, by adding exactly sampled midpoints, where the probability that they reach the target is non-negligible. Compared to a path of N equally spaced points, the algorithm achieves the same numerical accuracy N eff , while sampling only a small fraction of all points. Though this induces a statistical error, the latter is bounded for each bridge, allowing us to bound the total error rate by a number of our choice, say P tot error = 10 −6 . This leads to significant improvements in both memory and speed. For H = 0.33 and N eff = 2 32 , we need 5 000 times less CPU time and 10 000 times less memory than the classical Davies Harte algorithm. The gain grows for H = 0.25 and N eff = 2 42 to 3 · 10 5 for CPU and 10 6 for memory. We estimate our algorithmic complexity as C ABSec (N eff ) = O (ln N eff ) 3 , to be compared to Davies Harte which has complexity C DH (N ) = O (N ln N ). Decreasing P tot error results in a small increase in complexity, proportional to ln(1/P tot error ). Our current implementation is limited to the values of N eff given above, due to a loss of floating-point precision. Our algorithm can be adapted to other extreme events and arbitrary Gaussian processes. It enables one to numerically validate theoretical predictions that were hitherto inaccessible.

Probability density of the fractional Langevin equation with reflecting walls

“…The FBM simulations on the semi-infinite interval agree with previous perturbative and numerical results in the literature [48][49][50][51][52]61]. Specifically, the survival probability decays as S(t) ∼ t α/2−1 for long times, 2Kt α…”

Theory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles

“…Since X t is a Gaussian process, many observables can be calculated analytically. This is interesting, since one can access analytically, in an expansion in H − 1/2, most variables of interest for extremal statistics [453,452,454,455,456,457,458,459,460,461,462]. An example of such an observable is the maximum relative height of elastic interfaces in a random medium [463].…”

Functionals of fractional Brownian motion and the three arcsine laws

“…Details of this algorithm are given in App. D. Interestingly, for the first-passage time, recently an algorithm was introduced which grows as ln(N ) 3 , albeit accepting a small error probability [66,72,73], allowing for even more precise estimates.…”

“…In general these algorithms generate the full trajectory. If one is only interested in a specific observable, as the first-passage time, not all points need to be generated, allowing for tremendous gains both in memory usage and execution speed [66,72,73].…”