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

Abstract

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