  We characterize the small-time asymptotic behavior of the exit probability of
a L\'evy process out of a two-sided interval and of the law of its overshoot,
conditionally on the terminal value of the process. The asymptotic expansions
are given in the form of a first-order term and a precise computable error
bound. As an important application of these formulas, we develop a novel
adaptive discretization scheme for the Monte Carlo computation of functionals
of killed L\'evy processes with controlled bias. The considered functionals
appear in several domains of mathematical finance (e.g., structural credit risk
models, pricing of barrier options, and contingent convertible bonds) as well
as in natural sciences. The proposed algorithm works by adding discretization
points sampled from the L\'evy bridge density to the skeleton of the process
until the overall error for a given trajectory becomes smaller than the maximum
tolerance given by the user.

Probability;Computational Finance