On pathwise stochastic integration

In this article, we construct a mapping I: D[0, ∞)×D[0,∞)→D[0,∞) hat if (Xt) is a semimartingale on a probability space (Ω, F, P) with respect to a filtration (Ft) and if (ft) is an r.c.l.l. (Ft) adapted process, then I(ƒ.(ω), X. (ω))= ∫0.ƒ−dX(ω) a.s. s of significance when using stochastic integrals in statistical inference problems. Similar results on solutions to SDEs are also given.