Exact Asymptotics of Distributions of Integral Functionals of the Geometric Brownian Motion and Other Related Formulas

We prove results on exact asymptotics of the probabilities
$$ \mathbf P\biggl\{\int\limits_0^1e^{\varepsilon\xi(t)}\,dt>b\biggr\},\qquad \mathbf P\biggl\{\int\limits_0^1e^{|\varepsilon\xi(t)|}\,dt>b\biggr\},\qquad \varepsilon\to0, $$
where $b>1$, for two Gaussian processes $\xi(t)$, namely, a Wiener process and a Brownian bridge. We use the Laplace method for Gaussian measures in Banach spaces. Evaluation of constants is reduced to solving an extreme value problem for the rate function and studying the spectrum of a second-order differential operator of the Sturm–Liouville type with the use of Legendre functions.