On the closure in the Emery topology of semimartingale wealth-process sets

TL;DR

This paper investigates the properties of wealth-process sets in financial models, demonstrating that under no arbitrage conditions, all wealth processes are semimartingales and their set closure includes optimal processes in the Emery topology.

Contribution

It proves that wealth-process sets are semimartingales and their Emery topology closure contains all optimal wealth processes under no arbitrage of the first kind.

Findings

All wealth processes are semimartingales.

Closure of wealth-process set in Emery topology contains all optimal processes.

No arbitrage of the first kind ensures these properties.

Abstract

A wealth-process set is abstractly defined to consist of nonnegative c\`{a}dl\`{a}g processes containing a strictly positive semimartingale and satisfying an intuitive re-balancing property. Under the condition of absence of arbitrage of the first kind, it is established that all wealth processes are semimartingales and that the closure of the wealth-process set in the Emery topology contains all "optimal" wealth processes.