2 Citations (Scopus)

Abstract

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.

Original languageEnglish (US)
Pages (from-to)690-715
Number of pages26
JournalSIAM Journal on Financial Mathematics
Volume9
Issue number2
DOIs
StatePublished - Jan 1 2018

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Coherent Risk Measures
Continuous-time Markov Chain
Risk Evaluation
Markov processes
Risk Measures
Markov chain
Time Consistency
Generalise
Kolmogorov Equation
Multivalued Mapping
Differentiability
System of Ordinary Differential Equations
Markov Process
Continuous Time
Discrete-time
Generator
Concepts
Continuous-time Markov chain
Coherent risk measures
Costs

Cite this

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title = "Time-coherent risk measures for continuous-time markov chains",
abstract = "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.",
author = "{Dentcheva Ruszczynski}, Darinka and Andrzej Ruszczynski",
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Time-coherent risk measures for continuous-time markov chains. / Dentcheva Ruszczynski, Darinka; Ruszczynski, Andrzej.

In: SIAM Journal on Financial Mathematics, Vol. 9, No. 2, 01.01.2018, p. 690-715.

Research output: Contribution to journalArticle

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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.

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