Hyperamarts: Conditions for regularity of continuous parameter processes

The regularity of trajectories of continuous parameter process (X_t)_t∈R+ in terms of the convergence of sequence E(X_Tn) for monotone sequences (T_n) of stopping times is investigated. The following result for the discrete parameter case generalizes the convergence theorems for closed martingales: For an adapted sequence (X_n)_1≤n≤∞ of integrable random variables, lim X_n exists and is equal to X_∞ and (X_T) is uniformly integrable over the set of all extended stopping times T, if and only if lim E(X_Tn) = E(X_∞) for every increasing sequence (T_n) of extended simple stopping times converging to ∞. 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 (X_t)_t∈R+ with E{X_T} < ∞ for every bounded stopping time T is right continuous if lim E(X_Tn) = E(X_T) for every bounded stopping time T and every descending sequence (T_n) of bounded stopping times converging to T. Also, Riesz decomposition of a hyperamart is obtained.