In this work, we study a robust inventory management problem under demand and lead time uncertainty. Classical robust models assume that uncertain parameters are identically and independently distributed. However, in practical inventory management problems, correlations exist between demand and lead time. In our work, we propose a robust model with a joint uncertainty set that integrates correlations of the different sources of uncertainty. We study the pertinence of that model under different correlation assumptions (autocorrelated demand, correlated lead time and demand) and the impact on the solutions proposed.