# On the compensator in the Doob-Meyer decomposition of the Snell envelope

## TL;DR

This paper investigates the structure of the Snell envelope in optimal stopping problems, establishing conditions for its finite-variation part, linking it to the generator of the underlying process, and applying these results to control problems and smooth pasting.

## Contribution

It provides new insights into the compensator in the Doob-Meyer decomposition of the Snell envelope, especially in the Markovian setting, and connects it to stochastic control and smooth pasting conditions.

## Key findings

- Finite-variation part of the Snell envelope is absolutely continuous w.r.t. the decreasing part of the original process.
- Conditions identified for the value function to be in the domain of the extended generator.
- Dual problem of optimal stopping is characterized as a stochastic control problem with explicit control functions.

## Abstract

Let $G$ be a semimartingale, and $S$ its Snell envelope. Under the assumption that $G\in\mathcal{H}^1$, we show that the finite-variation part of $S$ is absolutely continuous with respect to the decreasing part of the finite-variation part of $G$. In the Markovian setting, this enables us to identify sufficient conditions for the value function of the optimal stopping problem to belong to the domain of the extended (martingale) generator of the underlying Markov process. We then show that the \textit{dual} of the optimal stopping problem is a stochastic control problem for a controlled Markov process, and the optimal control is characterised by a function belonging to the domain of the martingale generator. Finally, we give an application to the smooth pasting condition.