Representation of convex operators and their static and dynamic sandwich extensions

TL;DR

This paper investigates the dual representation and extension of monotone convex operators in stochastic optimization and finance, focusing on preserving properties like the sandwich condition and time-consistency across various temporal frameworks.

Contribution

It introduces new sandwich preserving extension theorems for monotone convex operators on subspaces of Lp, ensuring time-consistency in dynamic settings.

Findings

Established dual representation for convex operators on subspaces of Lp.

Proved sandwich preserving extension theorems for various time frameworks.

Ensured time-consistency of the extended operators.

Abstract

Monotone convex operators and time-consistent systems of operators appear naturally in stochastic optimization and mathematical finance in the context of pricing and risk measurement. We study the dual representation of a monotone convex \emph{operator} when its domain is defined on a subspace of $L_p$, with $p\in [1,\infty]$, and we prove a sandwich preserving extension theorem. These results are then applied to study systems of such operators defined only on subspaces. We propose various dynamic sandwich preserving extension results depending on the nature of time: finite discrete, countable discrete, and continuous. Of particular notice is the fact that the extensions obtained are time-consistent.