Stochastic dominated convergence theorem

If uniformly bounded predictable processes $H_n$ converge pointwise to $H$, then their integrals against a fixed semimartingale converge to $H\cdot X$ uniformly on compact intervals in probability. Localized domination gives corresponding extensions. This continuity permits a functional monotone-class argument from elementary predictable integrands to all bounded predictable integrands.

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