The Two-Parameter Brownian Bridge: Kolmogorov Inequalities and Upper and Lower Bounds for the Distribution of the Maximum
Cabana, Enrique M. ; Wschebor, Mario
Ann. Probab., Tome 10 (1982) no. 4, p. 289-302 / Harvested from Project Euclid
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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.
Publié le : 1982-05-14
Classification:  Two-parameter Brownian Bridge,  Kolmogorov inequalities,  distribution of the maximum,  heat equation,  60G99,  62G10 
@article{1176993858,
     author = {Cabana, Enrique M. and Wschebor, Mario},
     title = {The Two-Parameter Brownian Bridge: Kolmogorov Inequalities and Upper and Lower Bounds for the Distribution of the Maximum},
     journal = {Ann. Probab.},
     volume = {10},
     number = {4},
     year = {1982},
     pages = { 289-302},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1176993858}
}

Cabana, Enrique M.; Wschebor, Mario. The Two-Parameter Brownian Bridge: Kolmogorov Inequalities and Upper and Lower Bounds for the Distribution of the Maximum. Ann. Probab., Tome 10 (1982) no. 4, pp.  289-302. http://gdmltest.u-ga.fr/item/1176993858/