Time-Coherent Risk Measures for Continuous-Time Markov Chains

We propose an approach to risk evaluation of cost processes in continuous-time Markov chains. Our analysis is based on dual representation of coherent risk measures, differentiability concepts for multivalued mappings, and a concept of time coherence as refined 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 chain. We introduce the concept of a semiderivative of a risk multikernel and use it to generalize the concept of a generator of a Markov chain. Using these semiderivatives, we derive a system of ordinary differential equations that the risk evaluation must satisfy, which generalize the classical backward Kolmogorov equations for Markov processes. Furthermore, we discuss when such a system can be used to construct a dynamic risk measure. Additionally, we construct convergent discrete-time approximations to the continuous-time measures.