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A lower bound for boundary crossing probabilities of Brownian bridge/motion with trend

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  • Bischoff, Wolfgang
  • Hashorva, Enkelejd

Abstract

We show a lower bound for the boundary crossing probability P{[there exists]z[set membership, variant][0,1]:h(z)+B0(z)>u(z)} with B0 a Brownian bridge, h a trend function and u a boundary function. By that we get also a lower bound for the boundary crossing probability P{[there exists]z[set membership, variant][0,1]:h(z)+B0(z) [infinity]. The usual tools to obtain boundary crossing probabilities are finite dimensional approximations of the above probability. In this paper, however, we use the Cameron-Martin-Girsanov formula to obtain a bound of the above probability.

Suggested Citation

  • Bischoff, Wolfgang & Hashorva, Enkelejd, 2005. "A lower bound for boundary crossing probabilities of Brownian bridge/motion with trend," Statistics & Probability Letters, Elsevier, vol. 74(3), pages 265-271, October.
    • Handle: RePEc:eee:stapro:v:74:y:2005:i:3:p:265-271
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    References listed on IDEAS

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    1. Abundo, Mario, 2002. "Some conditional crossing results of Brownian motion over a piecewise-linear boundary," Statistics & Probability Letters, Elsevier, vol. 58(2), pages 131-145, June.
    2. Wolfgang Bischoff & Frank Miller & Enkelejd Hashorva & Jürg Hüsler, 2003. "Asymptotics of a Boundary Crossing Probability of a Brownian Bridge with General Trend," Methodology and Computing in Applied Probability, Springer, vol. 5(3), pages 271-287, September.
    3. Bischoff, Wolfgang & Hashorva, Enkelejd & Hüsler, Jürg & Miller, Frank, 2004. "On the power of the Kolmogorov test to detect the trend of a Brownian bridge with applications to a change-point problem in regression models," Statistics & Probability Letters, Elsevier, vol. 66(2), pages 105-115, January.
    4. Wolfgang Bischoff & Enkelejd Hashorva & Jürg Hüsler & Frank Miller, 2003. "Exact asymptotics for Boundary crossings of the brownian bridge with trend with application to the Kolmogorov test," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 55(4), pages 849-864, December.
    5. Janssen Arnold & Kunz Michael, 2002. "Global Extrapolations For Power Functions Of One-Sided Nonparametric Tests," Statistics & Risk Modeling, De Gruyter, vol. 20(1-4), pages 153-176, April.
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    Cited by:

    1. Enkelejd Hashorva, 2010. "Boundary Non-crossings of Brownian Pillow," Journal of Theoretical Probability, Springer, vol. 23(1), pages 193-208, March.
    2. Enkelejd Hashorva & Yuliya Mishura & Georgiy Shevchenko, 2021. "Boundary Non-crossing Probabilities of Gaussian Processes: Sharp Bounds and Asymptotics," Journal of Theoretical Probability, Springer, vol. 34(2), pages 728-754, June.
    3. Pingjin Deng, 2016. "Asymptotic of Non-Crossings probability of Additive Wiener Fields," Papers 1610.07131, arXiv.org.
    4. Wolfgang Bischoff & Andreas Gegg, 2016. "The Cameron–Martin Theorem for (p-)Slepian Processes," Journal of Theoretical Probability, Springer, vol. 29(2), pages 707-715, June.

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