On optimum strategies for minimizing the exponential moments of a loss function

We consider a general problem of minimizing the exponential moment of a given loss function, with an emphasis on the relation to the more common criterion of minimization the first moment of the same loss function. Our basic observation is about simple sufficient conditions for a strategy to be optimum in the exponential moment sense. This observation is useful and application examples are given. We also examine the asymptotic regime and investigate universal asymptotically optimum strategies in light of the aforementioned sufficient conditions.