Before revision:  To accommodate model ambiguity we introduce appropriate notions of history-consistent updating and of transition consistency for risk mappings on nested probability spaces .
After revision:  We confirm that the Markov property holds for the standard measures of risk used in practice such as Value at Risk and Average Value at Risk. We analyse the dual representation for convex Markovian risk mappings and a representation in terms of their acceptance sets. The Markov property is formulated in several equivalent versions including a strong version, opening up additional risk-sensitive optimisation problems such as optimal stopping with exercise lag and optimal prediction. We demonstrate how such problems can be reduced to a risk-sensitive optimal stopping problem with intermediate costs, and derive the dynamic programming equations for the latter. Finally, we show how our results can be extended to partially observable Markov processes .