Measurability of essential suprema in Snell envelope construction

Establish whether the measurability in the probability-law parameter of conditional expectations, as proved for bounded Borel functions under the topology of convergence in distribution, carries over to the essential suprema used in constructing Snell envelopes of American-style option payoffs under nondominated model uncertainty. Specifically, determine if the mapping that takes a probability measure to the essential supremum over stopping times of conditional expectations is measurable with respect to the Borel σ-algebra on the space of probability measures, without resorting to approximation via penalized backward stochastic differential equations.

Background

The paper develops an aggregator for a family of Snell envelopes to price and hedge American-style options under model uncertainty in a general semimartingale setting. A key technical hurdle is establishing measurability of objects that depend on both the underlying path and the probability law, in particular the Snell envelope which is defined via essential suprema of conditional expectations.

While measurability of conditional expectations with respect to the probability law is known in the literature, it is not immediately evident that this property extends to essential suprema arising in the Snell envelope construction. To circumvent this, the paper uses a penalization scheme and reflected BSDE approximations whose fixed-point construction yields the necessary measurability for the aggregator.

The authors point out explicitly that it is not clear whether the existing measurability results for conditional expectations directly extend to the essential suprema, which motivates this open question beyond their approximation-based approach.