  We introduce a notion of $k$th order stochastic monotonicity and duality that
allows one to unify the notion used in insurance mathematics (sometimes
refereed to as Siegmund's duality) for the study of ruin probability and the
duality responsible for the so-called put - call symmetries in option pricing.
Our general $k$th order duality can be financially interpreted as put - call
symmetry for powered options. The main objective of the present paper is to
develop an effective analytic approach to the analysis of duality leading to
the full characterization of $k$th order duality of Markov processes in terms
of their generators, which is new even for the well-studied case of put -call
symmetries.
