  We develop an approach to time-consistent risk evaluation of continuous-time
processes in Markov systems. Our analysis is based on dual representation of
coherent risk measures, differentiability concepts for multivalued mappings,
and a refined concept of time consistency. We prove that the risk measures are
defined by a family of risk evaluation functionals (transition risk mappings),
which depend on state, time, and the transition function. Their dual
representations are risk multikernels of the Markov system. We introduce the
concept of a semi-derivative of a risk multikernel and use it to generalize the
concept of a generator of a Markov process. Using these semi-derivatives, we
derive a system of ordinary differential equations that the risk evaluation
must satisfy, which generalize the classical backward Kolmogorov equations for
Markov processes. Additionally, we construct convergent discrete-time
approximations to the continuous-time risk measures.
