Closedness in the Semimartingale Topology For Spacesof Stochastic Integrals With Constrained Integrands

Let S be an Rd-valued semimartingale and ( n) a sequence of C-valued inte-grands, i.e., predictable, S-integrable processes taking values in some given closedset C(!, t) ⊆ Rd which may depend on the state ! and time t in a predictable way.Suppose that the stochastic integrals ( n · S) converge to X in the semimartingaletopology. We provide a necessary and sufficient condition (on S and C) that X canbe represented as stochastic integral with respect to S of some C-valued integrand,and we explain the relation to the sufficient conditions introduced earlier in [6], [20]and [21].[...]