Optional strong semimartingale inequalities for the strong Snell envelopes (Q6614289)

All the processes discussed below are assumed to be defined on a closed finite interval \([0,T]\) for some \(T>0\). Recall that a random process \(X\) belongs to the class \((D)\), if the family \(\{X_\tau: \tau<\infty\text{ is a stopping time}\}\) is uniformly integrable. Also recall that, for an optional semimartingale of class \((D)\), the smallest optional strong supermartingale which dominates \(X\) except on evanescent sets, is called strong Snell envelope.\N\NThe main result of the paper states that, for optional semimartingales \(X_1\) and \(X_2\) of class \((D)\) belonging to the space \(H^p\) for some \(p>1\) and their strong Snell envelopes \(Z_1\) and \(Z_2\), the following inequality holds: \(\|Z_1-Z_2\|_{H^p}\leq C_p \|X_1-X_2\|_{H^p}\), where \(C_p\) is an absolute constant. The proof is based on several characterizations of strong Snell envelopes, which are also obtained in the article.