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This is an overview of the work of the authors and their collaborators on the characterization of risk-sensitive costs and rewards in terms of an abstract Collatz–Wielandt formula and in case of rewards, also a controlled version of the Donsker–Varadhan formula. For the finite state and action case, this leads to useful linear and dynamic programming formulations for the reward maximization problem in the reducible case.

“Controlled” Versions of the Collatz–Wielandt and Donsker–Varadhan Formulae