The general problem of how to construct stochastic processes which are confined to stay in a predefined cone (in the one-dimensional but also multi-dimensional case also referred to as \emph{subordinators}) is of course known to be of great importance in the theory and a myriad of applications.\par But fuzzy stochastic processes are considered in this context for the first time in this paper:\par By first relating with each proper convex cone $C$ in $\R^{n}$ a certain cone of fuzzy vectors $C^*$ and subsequently using some specific Banach space techniques we have been able to produce as many pairs $(L^*_t, C^*)$ of fuzzy Łprocesses $L^*_t$ and cones $C^*$ of fuzzy vectors such that $L^*_t$ are $C^*$-$\,$subordinators.