# Optimal stopping without Snell envelopes

## TL;DR

This paper establishes the existence of optimal stopping times using elementary functional analysis, avoiding traditional Snell envelope methods, and introduces a duality framework involving martingales.

## Contribution

It provides the most general existence results for optimal stopping times by employing convex analysis and duality, sidestepping the need for Snell envelopes.

## Key findings

- Existence of optimal stopping times proved via convex analysis.
- Dual problem characterized by martingales dominating the reward.
- Generalized approach applicable to broader classes of problems.

## Abstract

This paper proves the existence of optimal stopping times via elementary functional analytic arguments. The problem is first relaxed into a convex optimization problem over a closed convex subset of the unit ball of the dual of a Banach space. The existence of optimal solutions then follows from the Banach--Alaoglu compactness theorem and the Krein--Millman theorem on extreme points of convex sets. This approach seems to give the most general existence results known to date. Applying convex duality to the relaxed problem gives a dual problem and optimality conditions in terms of martingales that dominate the reward process.