The Two-Parameter Brownian Bridge: Kolmogorov Inequalities and Upper and Lower Bounds for the Distribution of the Maximum

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.