A filtered version of the bipolar theorem of Brannath and Schachermayer

TL;DR

This paper extends the Bipolar Theorem to nonnegative cadlag supermartingales, introduces fork-convexity, and applies these results to characterize dual processes and admissible consumption in financial markets.

Contribution

It generalizes the Bipolar Theorem to a broader class of stochastic processes and introduces fork-convexity as a new concept in this context.

Findings

Extended Bipolar Theorem to cadlag supermartingales

Introduced fork-convexity as an analogue to convexity

Characterized dual processes and admissible consumption in finance

Abstract

We extend the Bipolar Theorem of Brannath and Schachermayer (1999) to the space of nonnegative cadlag supermartingales on a filtered probability space. We formulate the notion of fork-convexity as an analogue to convexity in this setting. As an intermediate step in the proof of our main result we establish a conditional version of the Bipolar theorem. In an application to mathematical finance we describe the structure of the set of dual processes of the utility maximization problem of Kramkov and Schachermayer (1999) and give a budget-constraint characterization of admissible consumption processes in an incomplete semimartingale market.