Fenchel-Moreau Conjugation Inequalities with Three Couplings and Application to Stochastic Bellman Equation

Given two couplings between "primal" and "dual" sets, we prove a general implication that relates an inequality involving "primal" sets to a reverse inequality involving the "dual" sets.% More precisely, let be given two "primal" sets PRIMAL, PRIMALBISand two "dual" sets DUAL, DUALBIS, together with two coupling functions (PRIMAL oversetcouplingleftrightarrow DUAL ) and (PRIMALBIS oversetcouplingbisleftrightarrow DUALBIS ). We define a new coupling (SumCouplingcouplingcouplingbis ) between the "primal" product set~PRIMAL × PRIMALBIS and the "dual" product set DUAL × DUALBIS. Then, we consider any bivariate function (kernel : PRIMAL × PRIMALBIS → barRR ) and univariate functions (fonctionprimal : PRIMAL → barRR ) and (fonctionprimalbis : PRIMALBIS → barRR ), all defined on the "primal" sets. We prove that (fonctionprimalnpprimal ≥ infprimalbis ∈ PRIMALBIS Bpkernelnpprimal, primalbis UppPlus fonctionprimalbisnpprimalbis ) ( Rightarrow SFMfonctionprimalcouplingnpdual ≤ infdualbis ∈ DUALBIS BpSFMkernelSumCouplingcouplingcouplingbisnpdual,dualbis UppPlus SFMfonctionprimalbis-couplingbisnpdualbis ), where we stress that the Fenchel-Moreau conjugates (SFMfonctionprimalcoupling ) and (SFMfonctionprimalbis-couplingbis) are not necessarily taken with the same coupling. We study the equality case, after having established the classical Fenchel inequality but with a general coupling. % We display several applications. We provide a new formula for the Fenchel-Moreau conjugate of a generalized inf-convolution. We obtain formulas with partial Fenchel-Moreau conjugates. Finally, we consider the Bellman equation in stochastic dynamic programming and we provide a "Bellman-like" equation for the Fenchel conjugates of the value functions.