Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-23T18:21:17.120Z Has data issue: false hasContentIssue false
  • English
  • Français

First Passage Time of Skew Brownian Motion

Part of: Stochastic analysis Stochastic processes

Published online by Cambridge University Press:  04 February 2016

Thilanka Appuhamillage  and
Daniel Sheldon
Thilanka Appuhamillage*
Affiliation:
Oregon State University
Daniel Sheldon*
Affiliation:
Oregon State University
*
∗ Postal address: Department of Mathematics, Oregon State University, Corvallis, OR 97331, USA. Email address: [email protected]
∗∗ Postal address: School of Electrical Engineering and Computer Science, Oregon State University, Corvallis, OR 97331, USA. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

Skew Brownian motionranked excursion heightfirst passage time

MSC classification

Primary: 60G99: None of the above, but in this section
Secondary: 60H99: None of the above, but in this section
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 3 , September 2012 , pp. 685 - 696
Copyright
© Applied Probability Trust 

Footnotes

Research supported in part by the grant CMG EAR0724865 from the National Science Foundation.

Research supported in part by the grant DBI-0905885 from the National Science Foundation.

References

Appuhamillage, T. and Sheldon, D. (2011). First passage time of skew Brownian motion. Preprint. Available at http://arxiv.org/abs/1008.2989v2.Google Scholar
Appuhamillage, T. A. et al. (2010). Solute transport across an interface: a Fickian theory for skewness in breakthrough curves. Water Resources Res. 46, 13 pp.CrossRefGoogle Scholar
Appuhamillage, T. et al. (2011). Occupation and local times for skew Brownian motion with applications to dispersion across an interface. Ann. Appl. Prob. 21, 183–214.Google Scholar
Berkowitz, B., Cortis, A., Dror, I. and Scher, H. (2009). Laboratory experiments on dispersive transport across interfaces: the role of flow direction. Water Resources Res. 45, 6 pp.CrossRefGoogle Scholar
Bhattacharya, R. N. and Waymire, E. C. (2009). Stochastic Processes with Applications (SIAM Classics Appl. Math. 61). Society for Industrial and Applied Mathematics, Philadelphia, PA.Google Scholar
Blackwell, P. G. (1997). Random diffusion models for animal movement. Ecological Modelling 100, 87–102.Google Scholar
Csáki, E. and Hu, Y. (2004). Invariance principles for ranked excursion lengths and heights. Electron. Commun. Prob. 9, 14–21.CrossRefGoogle Scholar
Dalziel, B. D., Morales, J. M. and Fryxell, J. M. (2008). Fitting probability distributions to animal movement trajectories: using artificial neural networks to link distance, resources, and memory. Amer. Naturalist 172, 248–258.CrossRefGoogle ScholarPubMed
Decamps, M., Goovaerts, M. and Schoutens, W. (2006). Asymmetric skew Bessel processes and their applications to finance. J. Comput. Appl. Math. 186, 130–147.Google Scholar
Fauchald, P. and Tveraa, T. (2003). Using first-passage time in the analysis of area-restricted search and habitat selection. Ecology 84, 282–288.CrossRefGoogle Scholar
Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. I, 3rd edn. John Wiley, New York.Google Scholar
Harada, S. J. (2011). Fundamental solution to the heat equation with a discontinuous diffusion coefficient and applications to skew Brownian motion and oceanography. , Oregon State University.Google Scholar
Harrison, J. M. and Shepp, L. A. (1981). On skew Brownian motion. Ann. Prob. 9, 309–313.Google Scholar
Itô, K. and McKean, H. P. Jr. (1963). Brownian motions on a half line. Illinois J. Math. 7, 181–231.Google Scholar
Le Gall, J.-F. (1984). One-dimensional stochastic differential equations involving the local times of the unknown process. In Stochastic Analysis and Applications (Swansea, 1983; Lecture Notes Math. 1095), Springer, Berlin, pp. 51–82.CrossRefGoogle Scholar
Lejay, A. and Martinez, M. (2006). A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients. Ann. Appl. Prob. 16, 107–139.Google Scholar
McKenzie, H. W., Lewis, M. A. and Merrill, E. H. (2009). First passage time analysis of animal movement and insights into the functional response. Bull. Math. Biol. 71, 107–129.Google Scholar
Ouknine, Y. (1990). Le “Skew-Brownian motion” et les processus qui en dérivent. Teor. Veroyat. Primen. 35, 173–179 (in French). English translation: Theory Prob. Appl. 35, 163-169.Google Scholar
Pinaud, D., Cherel, Y. and Weimerskirch, H. (2005). Effect of environmental variability on habitat selection, diet, provisioning behaviour and chick growth in yellow-nosed albatrosses. Marine Ecology Progr. Ser. 298, 295–304.Google Scholar
Pitman, J. and Yor, M. (2001). On the distribution of ranked heights of excursions of a Brownian bridge. Ann. Prob. 29, 361–384.Google Scholar
Polovina, J. J., Howell, E., Kobayashi, D. R. and Seki, M. P. (2001). The transition zone chlorophyll front, a dynamic global feature defining migration and forage habitat for marine resources. Progr. Oceanography 49, 469–483.Google Scholar
Ramirez, J. M. (2011). Multi-skewed Brownian motion and diffusion in layered media. Proc. Amer. Math. Soc. 139, 3739–3752.Google Scholar
Ramirez, J. M. et al. (2006). A generalized Taylor–Aris formula and skew diffusion. Multiscale Model. Simul. 5, 786–801.CrossRefGoogle Scholar
Ramirez, J. M. et al. (2008). A note on the theoretical foundations of particle tracking methods in heterogeneous porous media. Water Resources Res. 44, 5 pp.Google Scholar
Schultz, C. B. and Crone, E. E. (2001). Edge-mediated dispersal behavior in a prairie butterfly. Ecology 82, 1879–1892.CrossRefGoogle Scholar
Turchin, P. (1991). Translating foraging movements in heterogeneous environments into the spatial distribution of foragers. Ecology 72, 1253–1266.CrossRefGoogle Scholar
Turchin, P. (1996). Fractal analyses of animal movement: a critique. Ecology 77, 2086–2090.Google Scholar
Walsh, J. B. (1978). A diffusion with a discontinuous local time. Asterisque 52–53, 37–45.Google Scholar
Wiens, J. A. and Milne, B. T. (1989). Scaling of ‘landscapes’ in landscape ecology, or, landscape ecology from a beetle's perspective. Landscape Ecology 3, 87–96.CrossRefGoogle Scholar