Publication Cover
Stochastics
An International Journal of Probability and Stochastic Processes
Volume 96, 2024 - Issue 1
57
Views
0
CrossRef citations to date
0
Altmetric
Research Article

First-exit-time problems for two-dimensional Wiener and Ornstein–Uhlenbeck processes through time-varying ellipses

ORCID Icon, ORCID Icon, ORCID Icon & ORCID Icon

References

  • M. Abundo, On the first-passage times of certain Gaussian processes, and related asymptotics, Stoch. Anal. Appl. 39(4) (2021), pp. 712–727. https://doi.org/10.1080/07362994.2020.1843495.
  • L. Arnold, Stochastic Differential Equations: Theory and Applications, John Wiley & Sons Inc, New York London Sydney, 1974.
  • M.P. Atkinson and D.I. Singham, Multidimensional hitting time results for Brownian bridges with moving hyperplanar boundaries, Stat. Prob. Lett. 100 (2015), pp. 85–92. https://doi.org/10.1016/j.spl.2015.02.006.
  • E. Benedetto, L. Sacerdote, and C. Zucca, A first passage problem for a bivariate diffusion process: numerical solution with an application to neuroscience when the process is Gauss–Markov, J. Comput. Appl. Math. 242 (2013), pp. 41–52. https://doi.org/10.1016/j.cam.2012.10.014.
  • A.N. Borodin and P. Salminen, Handbook of Brownian Motion–Facts and Formulae, 2nd ed., Birkhäuser Verlag, Basel, 2002.
  • P.C. Bressloff, B.A. Earnshaw, and M.J. Ward, Diffusion of protein receptors on a cylindrical dendritic membrane with partially absorbing traps, SIAM J. Appl. Math. 68(5) (2008), pp. 1223–1246. https://doi.org/10.1137/070698373.
  • P.G. Buckholtz and M.T. Wasan, First passage probabilities of a two-dimensional Brownian motion in an anisotropic medium, Sankhya A 41 (1979. pp. 198–206. Available at https://www.jstor.org/stable/25050195.
  • A. Cheviakov, M.J. Ward, and R. Straube, An asymptotic analysis of the mean first passage time for narrow escape problems: part II: the sphere, Multiscale Model. Simul. 8(3) (2010), pp. 836–870. https://doi.org/10.1137/10078262.
  • C.-S. Chou and H.-J. Lin, Some properties of CIR processes, Stoch. Anal. Appl. 24(4) (2006), pp. 901–912. https://doi.org/10.1080/07362990600753643..
  • A. Comtet, F. Cornu, and G. Schehr, Last-passage time for linear diffusions and application to the emptying time of a box, J. Stat. Phys. 181(5) (2020), pp. 1565–1602. https://doi.org/10.1007/s10955-020-02637-6.
  • A. Dassios and L. Li, Explicit asymptotics on first passage times of diffusion processes, Adv. Appl. Prob. 52(2) (2020), pp. 681–704. https://doi.org/10.1017/apr.2020.13.
  • M. Deaconu and S. Herrmann, Hitting time for bessel-walk on moving spheres algorithm, Ann. Appl. Prob. 23(6) (2013), pp. 2259–2289. https://doi.org/10.1214/12-AAP900.
  • A. Di Crescenzo, V. Giorno, A.G. Nobile, and L.M. Ricciardi, On a symmetry-based constructive approach to probability densities for two-dimensional diffusion processes, J. Appl. Prob. 32(2) (1995), pp. 316–336. https://doi.org/10.2307/3215291.
  • E. Di Nardo and G. D'Onofrio, A cumulant approach for the first-passage-time problem of the feller square-root process, Appl. Math. Comput. 391 (2021), pp. 125707. https://doi.org/10.1016/j.amc.2020.125707.
  • M. Dominé and V. Pieper, First passage time distribution of a two-dimensional wiener process with drift, Prob. Engin. Inf. Sci. 7(4) (1993), pp. 545–555. https://doi.org/10.1017/S0269964800003120.
  • W. Feller, Two singular diffusion problems, Ann. Math. 54(1) (1951), pp. 173–182.
  • J.C. Fu and T.L. Wu, Boundary crossing probabilities for high-dimensional Brownian motion, J. Appl. Prob. 53(2) (2016), pp. 543–553. https://doi.org/10.1017/jpr.2016.19.
  • C. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Science, Springer-Verlag, Berlin Heidelberg, 2010.
  • L.T. Giorgini, W. Moon, and J.S. Wettlaufer, Analytical survival analysis of the Ornstein–Uhlenbeck process, J. Stat. Phys. 181(6) (2020), pp. 2404–2414. https://doi.org/10.1007/s10955-020-02669-y.
  • V. Giorno and A.G. Nobile, On the first-passage time problem for a Feller-type diffusion process, Mathematics 9(19) (2021), pp. 2470. https://doi.org/10.3390/math9192470.
  • V. Giorno and A.G. Nobile, Time-inhomogeneous Feller-type diffusion process with absorbing boundary condition, J. Stat. Phys. 183(3) (2021), pp. 45. https://doi.org/10.1007/s10955-021-02777-3.
  • V. Giorno, A.G. Nobile, L.M. Ricciardi, and L. Sacerdote, Some remarks on the Rayleigh process, J. Appl. Prob. 23(2) (1986), pp. 398–408. https://doi.org/10.2307/3214182.
  • V. Giorno, A. Di Crescenzo, A.G. Nobile, and L.M. Ricciardi, First-crossing time problems for diffusion processes in R2 through particular closed curves, in 6th European Young Statisticians Meeting, M. Hála and M. Malí, eds., Charles University, Prague, 1989, pp. 112–119.
  • V. Giorno, A.G. Nobile, and L.M. Ricciardi, On neuronal firing via specially confined diffusion processes, Sci. Math. Japon. 58(2) (2003), pp. 265–294.
  • P. Graczyk and T. Jakubowski, Exit times and Poisson kernels of the Ornstein–Uhlenbeck diffusion, Stoch. Models 24(2) (2008), pp. 314–337. https://doi.org/10.1080/15326340802009337.
  • I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, 8th ed., Elsevier/Academic Press, Amsterdam, 2015.
  • P. Grandits, Asymptotics of the hitting probability for a small sphere and a two dimensional Brownian motion with discontinuous anisotropic drift, Bernoulli 27(2) (2021), pp. 853–865. https://doi.org/10.3150/20-BEJ1257.
  • D.S. Grebenkov and A.T. Skvortsov, Mean first-passage time to a small absorbing target in an elongated planar domain, New J. Phys. 22(11) (2020), pp. 113024. https://doi.org/10.1088/1367-2630/abc91f.
  • D.S. Grebenkov, D. Holcman, and R. Metzler, Preface: new trends in first-passage methods and applications in the life sciences and engineering, J. Phys. A. Math. Theor. 53(19) (2020), pp. 190301. https://doi.org/10.1088/1751-8121/ab81d5.
  • H. Gzyl, Hitting spheres with Brownian motion revisited, Stat. Prob. Lett. 155 (2019), pp. 108565. https://doi.org/10.1016/j.spl.2019.108565.
  • Y. Hamana and H. Matsumoto, Hitting times to spheres of Brownian motions with and without drifts, Proc. Amer. Math. Soc. 144(12) (2016), pp. 5385–5396. https://doi.org/10.1090/proc/13136.
  • S. Karlin and H.W. Taylor, A Second Course in Stochastic Processes, Academic Press, New York, 1981.
  • S. Kou and H. Zhong, First-passage-times of two-dimensional Brownian motion, Adv. Appl. Prob. 48(4) (2016), pp. 1045–1060. https://doi.org/10.1017/apr.2016.64.
  • M. Lefebvre, First-passage problems for degenerate two-dimensional diffusion processes, Test 12(1) (2003), pp. 125–139. https://doi.org/10.1007/BF02595815.
  • M. Lefebvre, Similarity solutions of partial differential equations in probability, J. Prob. Stat. 2011 (2011), pp. 1–13. https://doi.org/10.1155/2011/689427.
  • M. Lefebvre, A first-passage-place problem for integrated diffusion processes, J. Appl. Prob. (2023), pp. 1–13. Published online. https://doi.org/10.1017/jpr.2023.19.
  • M. Lefebvre and R. Labib, Hitting lines and circles with diffusion processes, Aust. N. Z. J. Stat. 38(2) (1996), pp. 213–222. https://doi.org/10.1111/j.1467-842X.1996.tb00677.x.
  • N.A. Licata and S.W. Grill, The first-passage problem for diffusion through a cylindrical pore with sticky walls, Eur. Phys. J. E. 30(4) (2009), pp. 439–447. https://doi.org/10.1140/epje/i2009-10529-0.
  • A.E. Lindsay, R.T. Spoonmore, and J.C. Tzou, Hybrid asymptotic-numerical approach for estimating first-passage-time densities of the two-dimensional narrow capture problem, Phys. Rev. E. 94(4) (2016), pp. 042418. https://doi.org/10.1103/PhysRevE.94.042418.
  • M.M. Marchione and E. Orsingher, Hitting distribution of a correlated planar Brownian motion in a disk, Mathematics 10(4) (2022), pp. 536. https://doi.org/10.3390/math10040536.
  • R. Metzler, S. Redner, and G. Oshanin, First-passage Phenomena and Their Applications, World Scientific Publications, Singapore, 2014.
  • A. Metzler, On the first passage problem for correlated Brownian motion, Stat. Prob. Lett. 80(5-6) (2010), pp. 277–284. https://doi.org/10.1016/j.spl.2009.11.001.
  • M.N.P. Nilsson, The moving-eigenvalue method: hitting time for itô processes and moving boundaries, J. Phys. A. Math. Theor. 53(40) (2020), pp. 405201. https://doi.org/10.1088/1751-8121/ab9c59.
  • A.G. Nobile, L.M. Ricciardi, and L. Sacerdote, Exponential trends of Ornstein–Uhlenbeck First-Passage-Time densities, J. Appl. Prob. 22(2) (1985), pp. 360–369. https://doi.org/10.2307/3213864.
  • S. Pillay, M.J. Ward, A. Peirce, and T. Kolokolnikov, An asymptotic analysis of the mean first passage time for narrow escape problems: part I: two-dimensional domains, Multiscale Model. Simul. 8(3) (2010), pp. 803–835. https://doi.org/10.1137/090752511.
  • S. Redner, A Guide to First-Passage Processes, Cambridge University Press, Cambridge, 2001. https://doi.org/10.1017/CBO9780511606014.
  • L.M. Ricciardi, A. Di Crescenzo, V. Giorno, and A.G. Nobile, An outline of theoretical and algorithmic approaches to first passage time problems with applications to biological modeling, Math. Jpn. 50(2) (1999), pp. 247–322.
  • L. Sacerdote, M. Tamborrino, and C. Zucca, First passage times of two-dimensional correlated processes: analytical results for the wiener process and a numerical method for diffusion processes, J. Comput. Appl. Math. 296 (2016), pp. 275–292. https://doi.org/10.1016/j.cam.2015.09.033.
  • H. Shan, R. Moreno-Bote, and J. Drugowitsch, Family of closed-form solutions for two-dimensional correlated diffusion processes, Phys. Rev. E. 100(3) (2019), pp. 032132. https://doi.org/10.1103/PhysRevE.100.032132.
  • M.J. Simpson, D.J. VandenHeuvel, J.M. Wilson, S.W. McCue, and E.J. Carr, Mean exit time for diffusion on irregular domains, New J. Phys. 23(4) (2021), pp. 043030. https://doi.org/10.1088/1367-2630/abe60d.
  • F.G. Tricomi, Funzioni ipergeometriche confluenti, Monografie Matematiche a cura del Consiglio Nazionale delle Ricerche. Edizioni Cremonese, Roma, 1954.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.