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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 (, ). Of particular interest are those control problems associated to linear systems with Markov jump parameters (,,). 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 (). 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.