In this paper, we design a novel axiomatic approach to evaluating the joint risk of multiple insurance risks under dependence uncertainty. To be precise, we first establish a joint risk measure for non-negative multivariate risks, which we refer to as a (scalar) distortion joint risk measure. Then, we characterize it via a new set of axioms. Moreover, we introduce a new class of vector-valued distortion joint risk measures for non-negative multivariate risks and discuss their basic properties. Finally, comparisons with some existing vector-valued multivariate risk measures are made. It turns out that those vector-valued multivariate risk measures have forms of vector-valued distortion joint risk measures, respectively. This paper provides some relevant theoretical results about the evaluation of joint risk under dependence uncertainty.
