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We characterize the small-time asymptotic behavior of the exit probability ofa 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 expansionsare given in the form of a first-order term and a precise computable errorbound. As an important application of these formulas, we develop a noveladaptive discretization scheme for the Monte Carlo computation of functionalsof killed L\'evy processes with controlled bias. The considered functionalsappear in several domains of mathematical finance (e.g., structural credit riskmodels, pricing of barrier options, and contingent convertible bonds) as wellas in natural sciences. The proposed algorithm works by adding discretizationpoints sampled from the L\'evy bridge density to the skeleton of the processuntil the overall error for a given trajectory becomes smaller than the maximumtolerance given by the user.

Small-time asymptotics of stopped L\'evy bridges and simulation schemes with controlled bias