## The Annals of Probability

### 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.

#### Article information

Source
Ann. Probab., Volume 10, Number 2 (1982), 289-302.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176993858

Digital Object Identifier
doi:10.1214/aop/1176993858

Mathematical Reviews number (MathSciNet)
MR647505

Zentralblatt MATH identifier
0532.60072

JSTOR