We aim to analyse a Markovian discrete-time optimal stopping problem for a risk-averse decision maker under model ambiguity. In contrast to the analytic approach  based on transition risk mappings,  a probabilistic setting is introduced based on novel concepts of regular conditional risk mapping and Markov update rule. To accommodate model ambiguity we introduce appropriate notions of history-consistent updating and of transition consistency for risk mappings on nested probability spaces .
 In contrast to the analytic approach to risk for Markov chains based on transition risk mappings, we introduce a probabilistic setting based on a novel concept of regular conditional risk mapping with Markov update rule. 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 .
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We aim to analyse a Markovian discrete-time optimal stopping problem for a risk-averse decision maker under model ambiguity.
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to risk for Markov chains
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To accommodate model ambiguity we introduce appropriate notions of history-consistent updating and of transition consistency for risk mappings on nested probability spaces
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