Hyperamarts: Conditions for regularity of continuous parameter processes

The regularity of trajectories of continuous parameter process (Xt)t[set membership, variant]R+ in terms of the convergence of sequence E(XTn) for monotone sequences (Tn) of stopping times is investigated. The following result for the discrete parameter case generalizes the convergence theorems for closed martingales: For an adapted sequence (Xn)1<=n<=[infinity] of integrable random variables, lim Xn exists and is equal to X[infinity] and (XT) is uniformly integrable over the set of all extended stopping times T, if and only if lim E(XTn) = E(X[infinity]) for every increasing sequence (Tn) of extended simple stopping times converging to [infinity]. By applying these discrete parameter theorems, convergence theorems about continuous parameter processes are obtained. For example, it is shown that a progressive, optionally separable process (Xt)t[set membership, variant]R+ with E{XT} < [infinity] for every bounded stopping time T is right continuous if lim E(XTn) = E(XT) for every bounded stopping time T and every descending sequence (Tn) of bounded stopping times converging to T. Also, Riesz decomposition of a hyperamart is obtained.