TY - JOUR
T1 - Time-coherent risk measures for continuous-time markov chains
AU - Dentcheva, Darinka
AU - Ruszczynski, Andrzej
N1 - Publisher Copyright:
© 2018 Society for Industrial and Applied Mathematics.
PY - 2018
Y1 - 2018
N2 - 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.
AB - 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.
KW - Backward equations
KW - Discrete-time approximations
KW - Dynamic risk measures
KW - Risk multigenerators
KW - Risk multikernels
KW - Time consistency
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U2 - 10.1137/16M1063794
DO - 10.1137/16M1063794
M3 - Article
AN - SCOPUS:85049742190
VL - 9
SP - 690
EP - 715
JO - SIAM Journal on Financial Mathematics
JF - SIAM Journal on Financial Mathematics
IS - 2
ER -