On estimates and the asymptotic behavior of nonexit probabilities of a Wiener process to a moving boundary

In this paper the asymptotic behavior, as well as upper and lower bounds, is found for the probabilities $\mathsf P\{\sigma>T\}=\mathsf P\{|w_t|\leqslant f(t),0\leqslant t\leqslant T\}$, $\mathsf P\{\sigma>T\}=\mathsf P\{w_t\geqslant g(t),0\leqslant t\leqslant T\}$ for large classes of functions $f$ and $g$.
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