Closedness in the Semimartingale Topology For Spaces of Stochastic Integrals With Constrained Integrands

Abstract

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 closed set 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 semimartingale topology. We provide a necessary and sufficient condition (on S and C) that X can be 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]. The existence of such representations is equivalent to the closedness (in the semimartingale topology) of the space of stochastic integrals of C-valued integrands, which is crucial for the existence of solutions to most optimisation problems under trading constraints in mathematical finance. Moreover, we show that a predictably convex space of stochastic integrals is closed in the semimartingale topology if and only if it is a space of stochastic integrals of C-valued integrands, where each C(!, t) is convex.