The lower Snell envelope of smooth functions: an optional decomposition

Abstract

In this paper we provide general conditions under which the lower Snell envelope defined with respect to the family $\mathcal{M} $ of equivalent local-martingale probability measures of a semimartingale $S$ admits a decomposition as a stochastic integral with respect to $S$ and an optional process of finite variation. On the other hand, based on properties of predictable stopping times we establish a version of the classical backwards induction algorithm in optimal stopping for the non-linear super-additive expectation associated to $\mathcal{M} $. This result is of independent interest and we show how to apply it in order to systematically construct instances of the lower Snell envelope with no optional decomposition. Such ‘counterexamples’ strengths the scope of our conditions.