Abstract
The aim of this paper is to give upper and lower bounds for the probability density at $(u - z)$ of the position at time $(x, y) (x, y, z, u \in R^+)$ of a standard Wiener process with two-dimensional parameter $(x, y)$ with the requirement that it did not reach the barrier $u$ in the "past" $\{(x', y'): 0 \leq x' \leq x, 0 \leq y' \leq y\}$. The fundamental tools are Kolmogorov forward inequalities for the density and certain bounds for the behaviour of $p$ near the border.
Citation
Enrique M. Cabana. Mario Wschebor. "The Two-Parameter Brownian Bridge: Kolmogorov Inequalities and Upper and Lower Bounds for the Distribution of the Maximum." Ann. Probab. 10 (2) 289 - 302, May, 1982. https://doi.org/10.1214/aop/1176993858
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