Boundary non-crossings of Brownian pillow

TL;DR

This paper derives bounds and asymptotic behavior for the probability that a Brownian pillow with added functions does not cross a boundary, including solving a related minimization problem.

Contribution

It provides new bounds and asymptotic analysis for boundary non-crossing probabilities of Brownian pillows with added functions.

Findings

Established upper and lower bounds for boundary non-crossing probabilities.

Analyzed asymptotic behavior as the scaling factor tends to infinity.

Solved a related boundary minimization problem.

Abstract

Let B_0(s,t) be a Brownian pillow with continuous sample paths, and let h,u:[0,1]^2\to R be two measurable functions. In this paper we derive upper and lower bounds for the boundary non-crossing probability \psi(u;h):=P{B_0(s,t)+h(s,t) \le u(s,t), \forall s,t\in [0,1]}. Further we investigate the asymptotic behaviour of $\psi(u;\gamma h)$ with $\gamma$ tending to infinity, and solve a related minimisation problem.