Optimal stopping under model uncertainty and the regularity of lower Snell envelopes

Abstract

The analysis of American options in incomplete markets has motivated the development of robust versions of the classical Snell envelopes: The cost of superhedging an American option is characterized by the upper Snell envelope, while the infimum of the arbitrage prices is given by the lower Snell envelope. Lower Snell envelopes also appear in the problem of optimal stopping under model uncertainty. In this paper we focus on the lower Snell envelope. We construct a regular version of this stochastic process. To this end, we apply results due to Dellacherie and Lenglart on the regularization of stochastic processes and 𝒯-Systems.