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which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.Abstract In this work we give a comprehensive overview of the time consistency property of dynamic risk and performance measures, focusing on a the discrete time setup. The two key operational concepts used throughout are the notion of the LMmeasure and the notion of the update rule that, we believe, are the key tools for studying time consistency in a unified framework.Keywords Time consistency · Update rule · Dynamic LM-measure · Dynamic risk measure · Dynamic acceptability index · Measure of performance"The dynamic consistency axiom turns out to be the heart of the matter."

In the paper portfolio optimization over long run risk sensitive criterion is considered. It is assumed that economic factors which stimulate asset prices are ergodic but non necessarily uniformly ergodic. Solution to suitable Bellman equation using local span contraction with weighted norms is shown. The form of optimal strategy is presented and examples of market models satisfying imposed assumptions are shown.