On Lenglart's Theory of Meyer-sigma-fields and El Karoui's Theory of Optimal Stopping

TL;DR

This paper reviews and extends El Karoui's optimal stopping theory for processes measurable with respect to Meyer-$\sigma$-fields, introducing new path regularity and limit results, and clarifying a proof issue.

Contribution

It provides novel path regularity and limit results for Meyer measurable processes and clarifies a proof in El Karoui's 1981 work, enabling new applications.

Findings

Path regularity results for Meyer measurable processes

Limit results for Meyer-projections

Alternative approach to optimal stopping using stochastic representation

Abstract

We summarize the general results of El Karoui [1981] on optimal stopping problems for processes which are measurable with respect to Meyer-$\sigma$-fields. Meyer-$\sigma$-fields are due to Lenglart [1980] and include the optional and predictable $\sigma$-field as special cases. Novel contributions of our work are path regularity results for Meyer measurable processes and limit results for Meyer-projections. We will also clarify a minor issue in the proof of the optimality result in El Karoui [1981]. These extensions were inspired and needed for the proof of a stochastic representation theorem in Bank and Besslich [2018a]. As an application of this theorem, we provide an alternative approach to optimal stopping in the spirit of Bank and F{\"o}llmer [2003].