Publication at Faculty of Mathematics and Physics |

2009

We study an extremal behavior of stochastic integrals with respect to Brownian motion. As the main result we obtain an exact tail behavior of a supremum of these integrals taken over [0,h] with h}0 fixed, and a limiting distribution of the supremum on intervals [0,T].

We show how the limit distribution is connected to an asymptotic of maximally selected quasi-likelihood procedure that is used to detect changes at an unknown time in polynomial regress on models. In an application to global near-surface temperatures we demonstrate that the limit results presented in the paper perform well for real data sets.