  We propose a general framework to study last passage times, suprema and
drawdowns of a large class of stochastic processes. A central role in our
approach is played by processes of class Sigma. After investigating convergence
properties and a family of transformations that leave processes of class Sigma
invariant, we provide three general representation results. The first one
allows to recover a process of class Sigma from its final value and the last
time it visited the origin. In many situations this gives access to the
distribution of the last time a stochastic process hit a certain level or was
equal to its running maximum. It also leads to a formula recently discovered by
Madan, Roynette and Yor expressing put option prices in terms of last passage
times. Our second representation result is a stochastic integral representation
of certain functionals of processes of class Sigma, and the third one gives a
formula for their conditional expectations. From the latter one can deduce the
laws of a variety of interesting random variables such as running maxima,
drawdowns and maximum drawdowns of suitably stopped processes. As an
application we discuss the pricing and hedging of options that depend on the
running maximum of an underlying price process and are triggered when the
underlying price drops to a given level or alternatively, when the drawdown or
relative drawdown of the underlying price attains a given height.
