Dual spaces of cadlag processes

TL;DR

This paper characterizes the dual spaces of cadlag processes, extending classical results to broader classes and providing foundational tools for stochastic analysis and optimal control problems.

Contribution

It extends functional analytic duality results for cadlag processes, including class (D) and regular processes, with applications to stochastic analysis and control.

Findings

Dual of cadlag processes of class (D) characterized by optional measures

Extensions of Bismut's projection results for continuous processes

Provides existence and optimality conditions for stochastic control problems

Abstract

This article characterizes topological duals of spaces of cadlag processes. We obtain extensions of functional analytic results of Dellacherie and Meyer that underlie many fundamental results in stochastic analysis. In particular, we obtain a characterization of the dual of cadlag processes of class $(D)$ in terms of optional measures of essentially bounded variation. When specialized to regular processes, we find extensions of the main result of Bismut \cite{bis78} on projections of continuous processes. The dual characterizations yield existence results and optimality conditions for many fundamental problems from optimal stopping to singular stochastic control well beyond classical formulations.