I am reading up about Levy processes and keep seeing the words overshoot and undershoot in the context of fluctuation theory and optimal stopping. Would anyone be able to clarify what these mean?
Any help is greatly appreciated.
2026-03-27 07:14:53.1774595693
What is an overshoot of a Levy process?
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The overshoot of a Levy process is the value of the process when it first crosses a fixed barrier. Let $X$ be a Levy process and fix $x > 0$. Denote by $T_x = \inf\{t>0, X_t >x\}$. Then $X_{T_x}$ is the overshoot. The undershoot is the same object with $x<0$. It can be interesting to know "how big" can a Levy process overshoot (or undershoot).