Semimartingale Inequalities for The Snell Envelopes

Abstract

On the finite time interval [0,T] we consider the class of right continuous (with left limits) processes and for each element its Snell envelope, that is the smallest supermartingale bounding X from above. We improve some known properties of the Snell envelopes based on which we prove the main result of this article, namely the inequality where X 1 and X 2 are two arbitrary semimartingales belonging to the space H p P> 1Y 1 and Y 2 are their corresponding Snell envelopes and C p are absolute constants. The key step in proving this result is a bound of the distance in variation between the predictable components of the Snell envelopes Y l and Y 2