Abstract
Abstract On finite time interval [ 0 , T ] {[0,T]} , let X be an optional semimartingale of class ( D ) {(D)} , and Z its strong Snell envelope, which is the smallest optional strong supermartingale bounding X above (except on evanescent set). In this article, we provide the several characterizations of strong Snell envelopes to establish the main result which is the following inequality: ∥ Z 1 - Z 2 ∥ ℍ p ≤ C p ∥ X 1 - X 2 ∥ ℍ p , \|Z^{1}-Z^{2}\|_{\mathbb{H}^{p}}\leq C_{p}\|X^{1}-X^{2}\|_{\mathbb{H}^{p}}, where X 1 {X^{1}} and X 2 {X^{2}} are two optional semimartingales of class ( D ) {(D)} belonging to the space ℍ p {\mathbb{H}^{p}} ( p > 1 {p>1} ), Z 1 {Z^{1}} and Z 2 {Z^{2}} are their Snell envelopes, C p {C_{p}} is an absolute constant.
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