Closedness in the semimartingale topology for spaces of stochastic integrals with constrained integrands

Abstract

Let S be an ℝd-valued semimartingale and (ψ n ) a sequence of C-valued integrands, i.e. predictable, S-integrable processes taking values in some given closed set C(ω, t) ⊆ ℝd which may depend on the state ω and time t in a predictable way. Suppose that the stochastic integrals (ψ n·S) converge to X in the semimartingale topology. When can X be represented as a stochastic integral with respect to S of some C-valued integrand? We answer this with a necessary and sufficient condition (on S and C), and explain the relation to the sufficient conditions introduced earlier in (Czichowsky, Westray, Zheng, Convergence in the semimartingale topology and constrained portfolios, 2010; Mnif and Pham, Stochastic Process Appl 93:149-180, 2001; Pham, Ann Appl Probab 12:143-172, 2002). The existence of such representations is equivalent to the closedness (in the semimartingale topology) of the space of all stochastic integrals of C-valued integrands, which is crucial in mathematical finance for the existence of solutions to most optimisation problems under trading constraints. 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(ω, t) is convex. © 2011 Springer-Verlag Berlin Heidelberg.