Journal of Integral Equations and Applications

On Integral Equations Arising in the First-Passage Problem for Brownian Motion

Goran Peskir

Full-text: Open access

Article information

Source
J. Integral Equations Applications, Volume 14, Number 4 (2002), 397-423.

Dates
First available in Project Euclid: 5 June 2007

Permanent link to this document
https://projecteuclid.org/euclid.jiea/1181074930

Digital Object Identifier
doi:10.1216/jiea/1181074930

Mathematical Reviews number (MathSciNet)
MR1984752

Zentralblatt MATH identifier
1044.60076

Subjects
Primary: 60J65: Brownian motion [See also 58J65] 45D05: Volterra integral equations [See also 34A12] 60J60: Diffusion processes [See also 58J65]
Secondary: 45G15: Systems of nonlinear integral equations 45G10: Other nonlinear integral equations 45Q05: Inverse problems 45K05: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20]

Keywords
The (inverse)first-passage problem Brownian motion a curve (nonlinear) boundary a first-passage time Markov process the ChapmanKolmogorov equation Volterra integral equation (of the first and second kind) a system of nonlinear integral equations

Citation

Peskir, Goran. On Integral Equations Arising in the First-Passage Problem for Brownian Motion. J. Integral Equations Applications 14 (2002), no. 4, 397--423. doi:10.1216/jiea/1181074930. https://projecteuclid.org/euclid.jiea/1181074930


Export citation

References

  • D. André, Solution directe du problème résolu par M. Bertrand, C.R. Acad. Sci. Paris 105 (1887), 436-437.
  • L. Bachelier, Théorie de la spéculation, Ann. Sci. École Norm. Sup. 17 (1900), 21-86; English trans. Theory of speculation in The random character of stock market prices (P.H. Cootner, ed.), MIT Press, Cambridge, MA, 1964, pp. 17-78.
  • A. Buonocore, A.G. Nobile and L.M. Ricciardi, A new integral equation for the evaluation of first-passage time probability densities, Adv. in Appl. Probab. 19 (1987), 784-780.
  • S. Chapman, On the Brownian displacements and thermal diffusion of grains suspended in a nonuniform fluid, Proc. Roy. Soc. London Ser. A 119 (1928), 34-54.
  • J.L. Doob, Heuristic approach to the Kolmogorov-Smirnov theorems, Ann. Math. Statist. 20 (1949), 393-403.
  • J. Durbin, The first-passage density of a continuous Gaussian process to a general boundary, J. Appl. Probab. 22 (1985), 99-122.
  • --------, The first-passage density of the Brownian motion process to a curved boundary (with an appendix by D. Williams), J. Appl. Probab. 29 (1992), 291-304.
  • A. Einstein, Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen, Ann. Phys. 17 (1905), 549-560; English transl. On the motion of small particles suspended in liquids at rest required by the molecular-kinetic theory of heat in Einstein's miraculous year, Princeton Univ. Press, 1998, pp. 85-98.
  • B. Ferebee, The tangent approximation to one-sided Brownian exit densities, Z. Wahrsch. Verw. Gebiete 61 (1982), 309-326.
  • A.D. Fokker, Die mittlere Energie rotierender elektrischer Dipole imStrahlungsfeld, Ann. Phys. 43 (1914), 810-820.
  • R. Fortet, Les fonctions aléatoires du type Markoff associées à certaines équations linéaires aux dérivées partielles du type parabolique, J. Math Pures Appl. (9) 22 (1943), 177-243.
  • H. Hochstadt, Integral equations, John Wiley & Sons, New York, 1973.
  • K. Itô and H.P. McKean, Jr., Diffusion processes and their sample paths, Springer-Verlag, New York-Berlin, 1996, reprint of 1965 original.
  • A.N. Kolmogorov, Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung, Math. Ann. 104 (1931), 415-458; English transl. On analytical methods in probability theory in Selected works of A.N. Kolmogorov, Vol. II (A.N. Shiryayev, ed.), Kluwer Acad. Publ., Dordrecht, 1992, pp. 62-108.
  • --------, Zur Theorie der stetigen zufälligen Prozesse, Math. Ann. 108 (1933), 149-160; English transl. On the theory of continuous random processes in Selected works of A.N. Kolmogorov, Vol. II (A.N. Shiryayev, ed.), Kluwer Acad. Publ., Dordrecht, 1992, pp. 156-168.
  • P. Lévy, Sur certains processus stochastiques homogènes, Compositio Math. 7 (1939), 283-339.
  • S. Malmquist, On certain confidence contours for distribution functions, Ann. Math. Statist. 25 (1954), 523-533.
  • C. Park and S.R. Paranjape, Probabilities of Wiener paths crossing differentiable curves, Pacific J. Math. 53 (1974), 579-583.
  • C. Park and F.J. Schuurmann, Evaluations of barrier-crossing probabilities of Wiener paths, J. Appl. Probab. 13 (1976), 267-275.
  • G. Peskir, Limit at zero of the Brownian first-passage density, Research Report No. 420, Dept. Theoret. Statist. Aarhus (2001), 12 pp.; Probab. Theory Related Fields 124 (2002), 100-111.
  • M. Planck, Über einen Satz der statistischen Dynamik and seine Erweiterung in der Quantentheorie, Sitzungsber. Preuß. Akad. Wiss. 24 (1917), 324-341.
  • L.M. Ricciardi, L. Sacerdote and S. Sato, On an integral equation for first-passage-time probability densities, J. Appl. Probab. 21 (1984), 302-314.
  • E. Schrödinger, Zur Theorie der Fall- und Steigversuche an Teilchen mit Brownscher Bewegung, Physik. Z. 16 (1915), 289-295.
  • A.J.F. Siegert, On the first passage time probability problem, Phys. Rev. 81 (1951), 617-623.
  • M. v. Smoluchowski, Einige Beispiele Brown'scher Molekularbewegung unter Einfluss aüsserer Kräfte, Bull. Intern. Acad. Sc. Cracovie A (1913), 418-434.
  • --------, Notiz über die Berechnung der Brownschen Molekularbewegung bei der Ehrenhaft-Millikanschen Versuchsanordnung, Physik. Z. 16 (1915), 318-321.
  • V. Strassen, Almost sure behavior of sums of independent random variables and martingales, Proc. Fifth Berkeley Symp. Math. Statist. Probab. (Berkeley 1965/66) Vol. II, Part 1, Univ. California Press, Berkeley, CA, pp. 315-343.
  • C. Zucca, Analytical, numerical and Monte Carlo techniques for the study of the first passage times, Ph.D. Thesis, University of Milano, 2001.