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Stochastic barriers for the Wiener process and a mathematical model

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Martingale Theory in Harmonic Analysis and Banach Spaces

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 939))

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Abstract

Let {W(t), 0≤t<∞} be the standard Wiener process. The probabilities of the type P[sup0≤t≤TW(t)−f(t)≧0] have been extensively studied when f(t) is a deterministic function. This paper discusses the probabilities of the type P<Superscript>p0≤t≤TW(t)−[f(t)+X(t)]≧0</Superscript> when X(t) is a stochastic process. By taking a compound Poisson process as X(t), an interesting mathematical model is created.

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References

  1. J. A. Beekman and C. P. Fuelling, Refined distributions for a multi-risk stochastic process, Scand. Actuar. J., (1977), 175–183.

    Google Scholar 

  2. J. A. Beekman and C. P. Fuelling, A multi-risk stochastic process, Trans. Soc. Actuar. 30 (1979), 371–397.

    Google Scholar 

  3. J. L. Doob, Stochastic Processes, John Wiley & Sons, New York, 1953.

    MATH  Google Scholar 

  4. J. L. Doob, Heuristic approach to the Kolmogorov-Smirnov theorem, Ann. Math. Stat 20 (1949), 393–403.

    Article  MathSciNet  MATH  Google Scholar 

  5. J. Durbin, Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test, J. Appl. Prob. 8 (1971), 431–453.

    Article  MathSciNet  MATH  Google Scholar 

  6. W. Feller, An Introduction to Probability Theory and its Applications, Vol. 1, John Wiley, New York, 1957.

    MATH  Google Scholar 

  7. C. Park and S. R. Paranjape, Probabilities of Wiener paths crossing differentiable curves, Pacific J. Math. 50 (1974), 579–583.

    Article  MathSciNet  MATH  Google Scholar 

  8. C. Park and F. J. Schuurmann, Evaluations of barrier-crossing probabilities of Wiener paths, J. Appl. Prob. 13 (1976), 267–275.

    Article  MathSciNet  MATH  Google Scholar 

  9. C. Park and F. J. Schuurmann, Evaluations of absorption probabilities for the Wiener process on large intervals, J. Appl. Prob. 17 (1980), 363–372.

    Article  MathSciNet  MATH  Google Scholar 

  10. R. Pyke, The supremum and infimum of the Poisson process, Ann. Math. Stat., 30 (1959), 568–576.

    Article  MathSciNet  MATH  Google Scholar 

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Author notes

  1. Chull Park

    Present address: Department of Mathematics and Statistics, Miami University, 45056, Oxford, Ohio

Authors and Affiliations

    Authors
    1. Chull Park
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    Jia-Arng Chao Wojbor A. Woyczyński

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    © 1982 Springer-Verlag

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    Park, C. (1982). Stochastic barriers for the Wiener process and a mathematical model. In: Chao, JA., Woyczyński, W.A. (eds) Martingale Theory in Harmonic Analysis and Banach Spaces. Lecture Notes in Mathematics, vol 939. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0096264

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    • DOI: https://doi.org/10.1007/BFb0096264

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    • Publisher Name: Springer, Berlin, Heidelberg

    • Print ISBN: 978-3-540-11569-4

    • Online ISBN: 978-3-540-39284-2

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