A random variable (RV) X is given a mimimum selling price SU(X):=supx{x+EU(X–x)} (S) and a maximum buying price BP(X):=infx{x+EP(X–x)} (B) where U(·) and P(·) are appropriate functions. These prices are derived from considerations of stochastic optimization with recourse, and are called recourse certainty equivalents (RCEs) of X. Both RCEs compute the ‘value’ of an RV as an optimization problem, and both problems (S) and (B) have meaningful dual problems, stated in terms of the Csiszár φ-divergence Iφ(p, q) := Σni=1 qiφ(Pi/qi) a generalized entropy function, measuring the distance between RVs with probability vectors p and q. The RCE SU was studied elsewhere, and applied to production, investment and insurance problems. Here we study the RCE BP, and apply it to problems of inventory control (where the attitude towards risk determines the stock levels and order sizes) and optimal insurance coverage, a problem stated as a game between the insurance company (setting the premiums) and the buyer of insurance, maximizing the RCE of his coverage.
