Duality in convex problems of Bolza over functions of bounded variation

TL;DR

This paper explores convex Bolza problems over functions of bounded variation, providing dual representations, existence conditions, and optimality criteria within a conjugate duality framework, with applications to financial economics.

Contribution

It introduces a duality-based approach for convex Bolza problems over functions of bounded variation, extending classical results to more general measures and economic contexts.

Findings

Derived dual representation of the optimal value function

Established conditions for the existence of solutions

Formulated extended Hamiltonian optimality conditions

Abstract

This paper studies convex problems of Bolza in the conjugate duality framework of Rockafellar. We parameterize the problem by a general Borel measure which has direct economic interpretation in problems of financial economics. We derive a dual representation for the optimal value function in terms of continuous dual arcs and we give conditions for the existence of solutions. Combined with well-known results on problems of Bolza over absolutely continuous arcs, we obtain optimality conditions in terms of extended Hamiltonian conditions.