Risk-averse sequential decision problems with time-consistent stochastic dominance constraints

We discuss ways of comparison of two random sequences and their application in multistage stochastic optimization problems. While various comparisons of stochastic sequences have been proposed in the literature, their integration in a sequential decision problem is non-trivial and usually results in a time-inconsistent evaluations and inconsistent decisions. We propose a framework for constructing stochastic orderings that enable time-consistent comparisons at any stage of the decision process and ensure the dominance property of the optimal policy. We use the comparisons as constraints in multi-stage stochastic problems. Particular attention is paid to constraints based on stochastic dominance of the second-order imposed conditionally; it reflects risk-averse preferences and results in problems amenable to numerical treatment. We derive optimality condition for the new problems in a special case and establish relations to the expected utility theory. Additionally, we propose a numerical method for solving the new problems.