Convex integral functionals of regular processes

TL;DR

This paper develops dual representations for convex integral functionals on regular processes, enabling systematic analysis of stochastic optimization and control problems using Banach space techniques.

Contribution

It introduces dual representations for convex integral functionals on regular processes and applies them to stochastic optimization and control problems.

Findings

Dual representation of convex integral functionals on regular processes.

Identification of the dual space with optional Radon measures.

Application to stochastic optimization and maximum principles for control.

Abstract

This article gives dual representations for convex integral functionals on the linear space of regular processes. This space turns out to be a Banach space containing many more familiar classes of stochastic processes and its dual can be identified with the space of optional Radon measures with essentially bounded variation. Combined with classical Banach space techniques, our results allow for a systematic treatment of stochastic optimization problems over BV processes and, in particular, yields a maximum principle for a general class of singular stochastic control problems.