Optimal stopping with f -expectations: the irregular case

TL;DR

This paper studies optimal stopping problems under non-linear f-expectations without regularity assumptions, characterizing the value process via a Snell envelope and reflected BSDEs, with applications to American option pricing.

Contribution

It introduces a framework for irregular reward processes in optimal stopping with f-expectations, including a comparison theorem for irregular RBSDEs and a new characterization of the value process.

Findings

The value family can be aggregated by an optional process Y.

Y is characterized as the ^f-Snell envelope of .

A comparison theorem for irregular RBSDEs is established.

Abstract

We consider the optimal stopping problem with non-linear $f$-expectation (induced by a BSDE) without making any regularity assumptions on the reward process $\xi$. and with general filtration. We show that the value family can be aggregated by an optional process $Y$. We characterize the process $Y$ as the $\mathcal{E}^f$-Snell envelope of $\xi$. We also establish an infinitesimal characterization of the value process $Y$ in terms of a Reflected BSDE with $\xi$ as the obstacle. To do this, we first establish a comparison theorem for irregular RBSDEs. We give an application to the pricing of American options with irregular pay-off in an imperfect market model.