Abstract

The regularity of trajectories of continuous parameter process ( X t ) t∈ R + in terms of the convergence of sequence E( X T n ) 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 T n ) = 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 T n ) = 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.

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