Advertisement

Ukrainian Mathematical Journal

, Volume 51, Issue 6, pp 942–947 | Cite as

On certain exact relations for sojourn probabilities of a wiener process

  • V. A. Gasanenko
Brief Communications
  • 26 Downloads

Abstract

New exact relations are proved for the sojourn probability of a Wiener process between two time-de-pendent boundaries. The proof is based on the investigation of the heat-conduction equation in the domain determined by these functions-boundaries. The relations are given in the form of series.

Keywords

Brownian Motion Price Option Wiener Process Contemporary Problem Volterra Integral Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L. A. Shepp, “A first passage problem for the Wiener process,”Ann. Math. Statist.,38, 1912–1914 (1967).MathSciNetGoogle Scholar
  2. 2.
    V. S. Korolyuk,Boundary-Value Problems for Complicated Poisson Processes [in Russian], Naukova Dumka, Kiev (1975)Google Scholar
  3. 3.
    A. A. Novikov, “A martingale approach in problems on first crossing time of nonlinear boundaries,”Proc. Steklov Inst. Math.,4, 141–163 (1983).Google Scholar
  4. 4.
    M. Yor, “On some exponential functions of Brownian motion,”Adv. Appl. Probab.,24, 509–531 (1992).zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    N. Kunitomo and M. Ikeda, “Pricing options with curved boundaries,”Math. Finance,2, 275–297 (1992).zbMATHCrossRefGoogle Scholar
  6. 6.
    M. Teuken and M. Goovaerts, “Double boundary crossing result for the Brownian motion,”Scand. Actuar. J.,2, 139–150 (1994).Google Scholar
  7. 7.
    G. A. Grinberg, “On a possible method of approach to problems of the theory of heat conduction and diffusion, wave problems, and other similar problems in the presence of moving boundaries and on some other applications of this method,”Prikl. Mat. Mekh.,31, No. 2, 193–203 (1967).Google Scholar
  8. 8.
    A. N. Tikhonov, A. B. Vasil'eva, and A. T. Sveshnikov,Differential Equations [in Russian], Nauka Moscow (1985).Google Scholar
  9. 9.
    A. N. Tikhonov and A. A. Samarskii,Equations of Mathematical Physics [in Russian], Vysshaya Shkola, Moscow (1953).Google Scholar
  10. 10.
    A. Friedman,Partial Differential Equations of Parabolic Type [Russian translation], Mir, Moscow (1968).zbMATHGoogle Scholar

Copyright information

© Kluwer Academic/Plenum Publishers 2000

Authors and Affiliations

  • V. A. Gasanenko

There are no affiliations available

Personalised recommendations