Risk-sensitive Markov Decision Processes with Risk Constraints of Coherent Risk Measures in Fuzzy and Stochastic Environment

Risk-sensitive decision making with constraints of coherent risk measures is discussed in Markov decision processes. Risk-sensitive expected rewards under utility functions are approximated by weighted average value-at-risks, and risk constraints are described by coherent risk measures. In this paper, coherent risk measures are represented as weighted average value-at-risks with the best risk spectrum derived from decision maker's risk averse utility, and the risk spectrum can inherit the risk averse property of the decision maker's utility as weighting. By perception-based extension for fuzzy random variables, a dynamic portfolio model with coherent risk measures is introduced. To find feasible regions, firstly a dynamic risk-minimizing problem is discussed by mathematical programming Next a risk-sensitive reward maximization problem under the feasible coherent risk constraints is demonstrated. A few numerical examples are given to understand the obtained results.