Abstract:
We consider first-passage times of the standard Brownian motion over boundaries of order $c\sqrt {t}$. Our main result determines the tail behaviour of such first-passage times. This generalizes the well-known results obtained by Novikov and Uchiyama.
The work of A. Sakhanenko and V. Wachtel was supported by the Russian Foundation for Basic Research and German Research Foundation (DFG), project no. 20-51-12007. The work of A. Sakhanenko was also carried out in part within the framework of the state contract of the Sobolev Institute of Mathematics, project no. 0314-2019-0008.
Citation:
Denis E. Denisov, Günter Hinrichs, Alexander I. Sakhanenko, Vitali I. Wachtel, “Crossing an Asymptotically Square-Root Boundary by the Brownian Motion”, Branching Processes and Related Topics, Collected papers. On the occasion of the 75th birthday of Andrei Mikhailovich Zubkov and 70th birthday of Vladimir Alekseevich Vatutin, Trudy Mat. Inst. Steklova, 316, Steklov Math. Inst., Moscow, 2022, 113–128; Proc. Steklov Inst. Math., 316 (2022), 105–120