Stochastic dominance of sums of risks under dependence conditions

We provide conditions for the stochastic dominance comparisons of a risk $X$ and an associated risk $X+Z$, where $Z$ represents the uncertainty due to the environment and where $X$ and $Z$ can be dependent. The comparisons depend on both the copula $C$ between the distributions of $X$ and $Z$ and on the distribution of $Z$. We provide two different conditions for $C$ which represents new positive dependence properties. Regarding $Z$, we need some symmetry or asymmetry (skew) properties. Some illustrative examples are provided.