Distribution of the time at which a Brownian motion is maximal before its first-passage time
نویسندگان
چکیده
We calculate analytically the probability density P (tm) of the time tm at which a continuous-time Brownian motion (with and without drift) attains its maximum before passing through the origin for the first time. We also compute the joint probability density P (M, tm) of the maximum M and tm. In the driftless case, we find that P (tm) has power-law tails: P (tm) ∼ t −3/2 m for large tm and P (tm) ∼ t −1/2 m for small tm. In presence of a drift towards the origin, P (tm) decays exponentially for large tm. The results from numerical simulations are in excellent agreement with our analytical predictions.
منابع مشابه
The first-passage area for drifted Brownian motion and the moments of the Airy distribution
An exact expression for the distribution of the area swept out by a drifted Brownian motion till its first-passage time is derived. A study of the asymptotic behaviour confirms earlier conjectures and clarifies their range of validity. The analysis leads to a simple closed-form solution for the moments of the Airy distribution. PACS numbers: 02.50.-r, 05.40.-a, 05.40.Jc Short title: First-passa...
متن کاملOn the area under a continuous time Brownian motion till its first-passage time
The area swept out under a one-dimensional Brownian motion till its first-passage time is analysed using a Fokker-Planck technique. We obtain an exact expression for the area distribution for the zero drift case, and provide various asymptotic results for the non-zero drift case, emphasising the critical nature of the behaviour in the limit of vanishing drift. The results offer important insigh...
متن کاملFirst Passage Time of Skew Brownian Motion
Nearly fifty years after the introduction of skew Brownian motion by Itô and McKean (1963), the first passage time distribution remains unknown. In this paper, we first generalize results of Pitman and Yor (2001) and Csáki and Hu (2004) to derive formulae for the distribution of ranked excursion heights of skew Brownian motion, and then use this result to derive the first passage time distribut...
متن کاملFirst Passage Time Distribution for Anomalous Diffusion
We study the first passage time (FPT) problem in Levy type of anomalous diffusion. Using the recently formulated fractional Fokker-Planck equation, we obtain an analytic expression for the FPT distribution which, in the large passage time limit, is characterized by a universal power law. Contrasting this power law with the asymptotic FPT distribution from another type of anomalous diffusion exe...
متن کاملDistribution of the time at which the deviation of a Brownian motion is maximum before its first-passage time
We calculate analytically the probability density P (t m) of the time t m at which a continuous-time Brownian motion (with and without drift) attains its maximum before passing through the origin for the first time. We also compute the joint probability density P (M, t m) of the maximum M and t m. In the driftless case, we find that P (t m) has power-law tails: P (t m) ∼ t −3/2 m for large t m ...
متن کامل