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On certain exact relations for sojourn probabilities of a wiener process

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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.

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References

  1. L. A. Shepp, “A first passage problem for the Wiener process,”Ann. Math. Statist.,38, 1912–1914 (1967).

    MathSciNet  Google Scholar 

  2. V. S. Korolyuk,Boundary-Value Problems for Complicated Poisson Processes [in Russian], Naukova Dumka, Kiev (1975)

    Google Scholar 

  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. M. Yor, “On some exponential functions of Brownian motion,”Adv. Appl. Probab.,24, 509–531 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  5. N. Kunitomo and M. Ikeda, “Pricing options with curved boundaries,”Math. Finance,2, 275–297 (1992).

    Article  MATH  Google Scholar 

  6. M. Teuken and M. Goovaerts, “Double boundary crossing result for the Brownian motion,”Scand. Actuar. J.,2, 139–150 (1994).

    Google Scholar 

  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. A. N. Tikhonov, A. B. Vasil'eva, and A. T. Sveshnikov,Differential Equations [in Russian], Nauka Moscow (1985).

    Google Scholar 

  9. A. N. Tikhonov and A. A. Samarskii,Equations of Mathematical Physics [in Russian], Vysshaya Shkola, Moscow (1953).

    Google Scholar 

  10. A. Friedman,Partial Differential Equations of Parabolic Type [Russian translation], Mir, Moscow (1968).

    MATH  Google Scholar 

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Authors
  1. V. A. Gasanenko
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Additional information

Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 6, pp. 842–846, June, 1999

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Gasanenko, V.A. On certain exact relations for sojourn probabilities of a wiener process. Ukr Math J 51, 942–947 (1999). https://doi.org/10.1007/BF02591981

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  • DOI: https://doi.org/10.1007/BF02591981

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