In this survey, we show that various stochastic optimization problems arising in
option theory, in dynamical allocation problems, and in the microeconomic theory
of intertemporal consumption choice can all be reduced to the same problem of
representing a given stochastic process in terms of running maxima of another
process. We describe recent results of Bank and El Karoui (2002) on the general
stochastic representation problem, derive results in closed form for Lévy processes
and diffusions, present an algorithm for explicit computations, and discuss some
applications.