Conditional Coherent and Convex Risk Measures Under Uncertainty

In this paper, we take a new perspective to describe the model uncertainty, and thus propose two new classes of risk measures under model uncertainty. To be precise, we use an auxiliary random variable to describe model uncertainty. By proposing new sets of axioms under model uncertainty, we axiomatically introduce and characterize conditional coherent and convex risk measures under a random environment, respectively. As examples, we also discuss the connections of the introduced conditional coherent risk measures under random environments with two existing risk measures. This paper mainly gives some theoretical results, and it is expected to make meaningful complement to the study of coherent and convex risk measures under model uncertainty.