Duality in convex stochastic optimization

TL;DR

This paper advances the theory of convex stochastic optimization by deriving explicit dual problems with two dual variables, establishing conditions for primal solutions and no duality gap without traditional assumptions, and extending applications to finance and control.

Contribution

It introduces a novel duality framework with explicit dual variables, broadening the scope of stochastic optimization and its applications in finance and control.

Findings

Explicit dual problem with two dual variables derived

Existence of primal solutions established without compactness assumptions

No duality gap proven under relaxed conditions

Abstract

This paper studies duality and optimality conditions in general convex stochastic optimization problems introduced by Rockafellar and Wets in 1976. We derive an explicit dual problem in terms of two dual variables, one of which is the shadow price of information while the other one gives the marginal cost of a perturbation much like in classical Lagrangian duality. Existence of primal solutions and the absence of duality gap are obtained without compactness or boundedness assumptions. In the context of financial mathematics, the relaxed assumptions are satisfied under the well-known no-arbitrage condition and the reasonable asymptotic elasticity condition of the utility function. We extend classical portfolio optimization duality theory to problems of optimal semi-static hedging. Besides financial mathematics, we obtain several new frameworks in stochastic programming and stochastic optimal control.