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We review the question of the extreme values attained by a random process. Werelate it to level crossings either to one boundary (first-passage problems)and two boundaries (escape problems). The extremes studied are the maximum, theminimum, the maximum absolute value and the range or span. We specialize indiffusion processes and present detailed results for the Wiener and Fellerprocesses.

Extreme values and the level-crossing problem. An application to the Feller process