Duality and equilibrium prices in economics of uncertainty

A random variable (RV) X is given aminimum selling price <Equation ID="E1"> <EquationSource Format="TEX"> $$S_U \left( X \right):=\mathop {\sup }\limits_x \left\{ {x + EU\left( {X - x} \right)} \right\}$$ </EquationSource> </Equation> and amaximum buying price <Equation ID="E2"> <EquationSource Format="TEX"> $$B_p \left( X \right):=\mathop {\inf }\limits_x \left\{ {x + EP\left( {X - x} \right)} \right\}$$ </EquationSource> </Equation> whereU(·) andP(·) are appropriate functions. These prices are derived from considerations ofstochastic optimization with recourse, and are calledrecourse certainty equivalents (RCE's) of X. Both RCE's compute the “value” of a RV as an optimization problem, and both problems (S) and (B) have meaningful dual problems, stated in terms of theCsiszár φ-divergence <Equation ID="E3"> <EquationSource Format="TEX"> $$I_\phi \left( {p,q} \right):=\sum\limits_{i=1}^n {q_i \phi \left( {\frac{{p_i }}{{q_i }}} \right)} $$ </EquationSource> </Equation> a generalized entropy function, measuring the distance between RV's with probability vectors p and q. The RCES <Subscript>U</Subscript> was studied elsewhere, and applied to production, investment and insurance problems. Here we study the RCEB <Subscript>P</Subscript>, and apply it to problems ofinventory control (where the attitude towards risk determines the stock levels and order sizes) andoptimal 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. Copyright Physica-Verlag 1997