A new concept of risk sensitivity is given which unveils a wide class of functions not detected as risk sensitive in the classical scenario. Benefiting from the broadening of this class beyond the classical real-valued convex format, the concept allows, for instance, to further inroads concerning risk sensitive optimal control problems - as did Jacobson (1973) for the so-called LEQG control problem ([12], [15]). Of particular interest are those control problems associated to linear systems with Markov jump parameters ([2],[3],[7]). Focusing the niche of the standard risk sensitive functions, we study - via a risk measure criteria - the connections between the two concepts. It turns out that both have a same risk measure whenever the original cost r.v. of interest is additive - which is the case of ([12]). Due to its richer structure, the correspondence between the cost r.v. of original interest and its risk sensitive version, as presented herein, typically does not exist as a function. Nonetheless, in a probabilistic sense, a characterization by means of a family of convex functions is made, which parallels the risk sensitive standard case.