  For a multivariate random walk with i.i.d. jumps satisfying the Cramer moment condition and having a mean vector with at least one negative component, we derive the exact asymptotics of the probability of ever hitting the positive orthant that is being translated to infinity along a fixed vector with positive components. This problem is motivated by and extends results from a paper by F. Avram et al. (2008) on a two-dimensional risk process. Our approach combines the large deviation techniques from a recent series of papers by A. Borovkov and A. Mogulskii with new auxiliary constructions, which enable us to extend their results on hitting remote sets with smooth boundaries to the case of boundaries with a "corner" at the "most probable hitting point". We also discuss how our results can be extended to the case of more general target sets. 
The exact asymptotics of the large deviation probabilities in the   multivariate boundary crossing problem