Duality between Probability and Optimization

Following the theory of idempotent measures of Maslov, a formalism analogous to probability calculus is obtained for optimization by replacing the classical structure of real numbers (R;+; ) by the idempotent semield obtained by endowing the set R [ f+1g with the \min" and \+" operations. To the probability of an event corresponds the cost of a set of decisions. To random variables correspond decision variables. Weak convergence, tightness and limit theorems of probability have an optimization counterpart which is useful to approximate Hamilton Jacobi Bellman (HJB) equation and to obtain asymptotics for this equation. The introduction of tightness for cost measures and its consequences is the main contribution of this paper. The link between the weak convergence and the epigraph convergence used in convex analysis is done. The Cramer transform used in the large deviation literature is de ned as the composition of the Laplace transform by the logarithm by the Fenchel transform. It transforms convolution into inf-convolution. Probabilistic results about processes with independent increments are then transformed into similar results on dynamic programming equations. Cramer transform gives new insight on the Hopf method used to compute explicit solutions of some HJB equations. It also explains the limit theorems obtained directly as the image of the classic limit theorems of probability. Bibliographic notes are given at the end of the paper.