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Boundary Non-crossings of Brownian Pillow

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  • Enkelejd Hashorva

    (University of Bern)

Abstract

Let B 0(s,t) be a Brownian pillow with continuous sample paths, and let h,u:[0,1]2→ℝ be two measurable functions. In this paper we derive upper and lower bounds for the boundary non-crossing probability $$\psi(u;h):=\mathbf{P}\big\{B_{0}(s,t)+h(s,t)\leq u(s,t),\forall s,t\in[0,1]\big\}.$$ Further we investigate the asymptotic behaviour of ψ(u;γ h) with γ tending to ∞ and solve a related minimisation problem.

Suggested Citation

  • Enkelejd Hashorva, 2010. "Boundary Non-crossings of Brownian Pillow," Journal of Theoretical Probability, Springer, vol. 23(1), pages 193-208, March.
    • Handle: RePEc:spr:jotpro:v:23:y:2010:i:1:d:10.1007_s10959-008-0191-5
    DOI: 10.1007/s10959-008-0191-5
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    References listed on IDEAS

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    1. 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.
    2. Wolfgang Bischoff & Enkelejd Hashorva & Jürg Hüsler & Frank Miller, 2005. "Analysis of a change-point regression problem in quality control by partial sums processes and Kolmogorov type tests," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 62(1), pages 85-98, September.
    3. Koning, Alex J. & Protasov, Vladimir, 2003. "Tail behaviour of Gaussian processes with applications to the Brownian pillow," Journal of Multivariate Analysis, Elsevier, vol. 87(2), pages 370-397, November.
    4. Enkelejd Hashorva, 2005. "Asymptotics and Bounds for Multivariate Gaussian Tails," Journal of Theoretical Probability, Springer, vol. 18(1), pages 79-97, January.
    5. 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.
    6. Csáki, Endre & Khoshnevisan, Davar & Shi, Zhan, 2000. "Boundary crossings and the distribution function of the maximum of Brownian sheet," Stochastic Processes and their Applications, Elsevier, vol. 90(1), pages 1-18, November.
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    Cited by:

    1. 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.

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