First 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 (2011) and Csáki and Hu (2004) to derive formulae for the distribution of ranked excursion heights of skew Brownian motion, and then use these results to derive the first passage time distribution.

[1]  J. Ramírez.  Multi-skewed Brownian motion and diffusion in layered media , 2011 .

[2]  S. Harada.  Fundamental solution to the heat equation with a discontinuous diffusion coefficient with applications to Skew Brownian Motion and oceanography , 2011 .

[3]  Edward C. Waymire,et al.  Corrections for "Occupation and local times for skew Brownian motion with applications to dispersion across an interface" , 2010, 1009.5410.

[4]  D. Sheldon,et al.  Ranked Excursion Heights and First Passage Time of Skew Brownian Motion , 2010 .

[5]  Brian D. Wood,et al.  Solute transport across an interface: A Fickian theory for skewness in breakthrough curves , 2010 .

[6]  M. Lewis,et al.  First Passage Time Analysis of Animal Movement and Insights into the Functional Response , 2009, Bulletin of mathematical biology.

[7]  Brian Berkowitz,et al.  Laboratory experiments on dispersive transport across interfaces: The role of flow direction , 2008 .

[8]  Benjamin D. Dalziel,et al.  Fitting Probability Distributions to Animal Movement Trajectories: Using Artificial Neural Networks to Link Distance, Resources, and Memory , 2008, The American Naturalist.

[9]  Edward C. Waymire,et al.  A note on the theoretical foundations of particle tracking methods in heterogeneous porous media , 2008 .

[10]  A. Lejay,et al.  A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients , 2006, math/0603214.

[11]  W. Schoutens,et al.  Asymmetric skew Bessel processes and their applications to finance , 2006 .

[12]  Brian Wood,et al.  A Generalized Taylor-Aris Formula and Skew Diffusion , 2006, Multiscale Model. Simul..

[13]  H. Weimerskirch,et al.  Effect of environmental variability on habitat selection, diet, provisioning behaviour and chick growth in yellow-nosed albatrosses , 2005 .

[14]  Yueyun Hu,et al.  Invariance principles for ranked excursion lengths and heights. , 2004 .

[15]  Bruce T. Milne,et al.  Scaling of ‘landscapes’ in landscape ecology, or, landscape ecology from a beetle's perspective , 1989, Landscape Ecology.

[16]  Torkild Tveraa,et al.  USING FIRST‐PASSAGE TIME IN THE ANALYSIS OF AREA‐RESTRICTED SEARCH AND HABITAT SELECTION , 2003 .

[17]  Elizabeth E. Crone,et al.  EDGE-MEDIATED DISPERSAL BEHAVIOR IN A PRAIRIE BUTTERFLY , 2001 .

[18]  J. Pitman,et al.  On the distribution of ranked heights of excursions of a Brownian bridge , 2001 .

[19]  D. Kobayashi,et al.  The transition zone chlorophyll front, a dynamic global feature defining migration and forage habitat for marine resources , 2001 .

[20]  Paul G. Blackwell,et al.  Random diffusion models for animal movement , 1997 .

[21]  Charles M. Grinstead,et al.  Introduction to probability , 1999, Statistics for the Behavioural Sciences.

[22]  Peter Turchin,et al.  Fractal Analyses of Animal Movement: A Critique , 1996 .

[23]  Peter Turchin,et al.  Translating Foraging Movements in Heterogeneous Environments into the Spatial Distribution of Foragers , 1991 .

[24]  R. Bhattacharya,et al.  Stochastic processes with applications , 1990 .

[25]  Jim Pitman,et al.  On Walsh's Brownian motions , 1989 .

[26]  J. Gall,et al.  One — dimensional stochastic differential equations involving the local times of the unknown process , 1984 .

[27]  J. Harrison,et al.  On Skew Brownian Motion , 1981 .

[28]  Kiyosi Itô,et al.  Brownian motions on a half line , 1963 .

[29]  C. P. Hubbard,et al.  A Note on the , 1951 .

[30]  W. Feller.  An Introduction to Probability Theory and Its Applications , 1959 .