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Boundary Non-crossings of Brownian Pillow

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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.

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  1. Department of Mathematical Statistics and Actuarial Science, University of Bern, Sidlerstrasse 5, 3012, Bern, Switzerland

    Enkelejd Hashorva

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  1. Enkelejd Hashorva
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Hashorva, E. Boundary Non-crossings of Brownian Pillow. J Theor Probab 23, 193–208 (2010). https://doi.org/10.1007/s10959-008-0191-5

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  • DOI: https://doi.org/10.1007/s10959-008-0191-5

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