Time-inconsistent multistage stochastic programs: martingale bounds

Abstract. It is well known that multistage programs, which maximize expectation or expected utility, allow a dynamic programming formulation, and that other objectives destroy the dynamic programming character of the problem. This paper considers a risk measure at the final stage of a multistage stochastic program. Although these problems are not time consistent, it is shown that optimal decisions evolve as a martingale. A verification theorem is provided, which characterizes optimal decisions by enveloping sub- and supermartingales. To obtain these characterizations the idea of a constant risk profile has to be given up and instead a risk profile, which varies over time, has to be accepted. The basis of the analysis is a new decomposition theorem for risk measures, which is able to recover the genuine risk measure by measuring risk conditionally.


University of Vienna



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